ABCF->ab-angle a

Average Error: 52.4 → 42.1

Time: 1.2min

Precision: binary64

Cost: 40780

Math

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

↓

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_3 := \sqrt{A + \left(C + t_1\right)}\\ t_4 := -\sqrt{2 \cdot \left(t_1 + \left(A + C\right)\right)}\\ t_5 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;B \leq -0.49:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{t_2}\right) \cdot t_4}{t_2}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{{\left({\left(t_2 \cdot F\right)}^{0.25}\right)}^{2} \cdot t_4}{t_2}\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{t_5 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_5}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* A C) 4.0)))
        (t_1 (hypot B (- A C)))
        (t_2 (fma -4.0 (* A C) (* B B)))
        (t_3 (sqrt (+ A (+ C t_1))))
        (t_4 (- (sqrt (* 2.0 (+ t_1 (+ A C))))))
        (t_5 (fma B B (* C (* -4.0 A)))))
   (if (<= B -0.49)
     (/ (* (* (sqrt F) (sqrt t_2)) t_4) t_2)
     (if (<= B -2.1e-88)
       (/
        (- (sqrt (* 2.0 (* t_5 (* F (fma 2.0 A (* -0.5 (/ (* B B) C))))))))
        t_5)
       (if (<= B -2.15e-177)
         (/ (* (pow (pow (* t_2 F) 0.25) 2.0) t_4) t_2)
         (if (<= B -3.4e-270)
           (/ (- (sqrt (* 2.0 (* t_5 (* F (* A 2.0)))))) t_5)
           (if (<= B 8e-293)
             (* (sqrt (* t_5 (* 2.0 (* 2.0 (* C F))))) (/ -1.0 t_5))
             (if (<= B 4.8e-268)
               (/ (* t_3 (- (sqrt (* 2.0 (* -4.0 (* A (* C F))))))) t_0)
               (if (<= B 4.9e-110)
                 (/
                  (-
                   (sqrt
                    (* 2.0 (* t_5 (* F (fma 2.0 A (* -0.5 (/ B (/ C B)))))))))
                  t_5)
                 (if (<= B 1.25e+74)
                   (/
                    (* t_3 (- (sqrt (* 2.0 (* F (fma B B (* -4.0 (* A C))))))))
                    t_0)
                   (* (sqrt 2.0) (- (sqrt (/ F B))))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - ((A * C) * 4.0);
	double t_1 = hypot(B, (A - C));
	double t_2 = fma(-4.0, (A * C), (B * B));
	double t_3 = sqrt((A + (C + t_1)));
	double t_4 = -sqrt((2.0 * (t_1 + (A + C))));
	double t_5 = fma(B, B, (C * (-4.0 * A)));
	double tmp;
	if (B <= -0.49) {
		tmp = ((sqrt(F) * sqrt(t_2)) * t_4) / t_2;
	} else if (B <= -2.1e-88) {
		tmp = -sqrt((2.0 * (t_5 * (F * fma(2.0, A, (-0.5 * ((B * B) / C))))))) / t_5;
	} else if (B <= -2.15e-177) {
		tmp = (pow(pow((t_2 * F), 0.25), 2.0) * t_4) / t_2;
	} else if (B <= -3.4e-270) {
		tmp = -sqrt((2.0 * (t_5 * (F * (A * 2.0))))) / t_5;
	} else if (B <= 8e-293) {
		tmp = sqrt((t_5 * (2.0 * (2.0 * (C * F))))) * (-1.0 / t_5);
	} else if (B <= 4.8e-268) {
		tmp = (t_3 * -sqrt((2.0 * (-4.0 * (A * (C * F)))))) / t_0;
	} else if (B <= 4.9e-110) {
		tmp = -sqrt((2.0 * (t_5 * (F * fma(2.0, A, (-0.5 * (B / (C / B)))))))) / t_5;
	} else if (B <= 1.25e+74) {
		tmp = (t_3 * -sqrt((2.0 * (F * fma(B, B, (-4.0 * (A * C))))))) / t_0;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0))
	t_1 = hypot(B, Float64(A - C))
	t_2 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_3 = sqrt(Float64(A + Float64(C + t_1)))
	t_4 = Float64(-sqrt(Float64(2.0 * Float64(t_1 + Float64(A + C)))))
	t_5 = fma(B, B, Float64(C * Float64(-4.0 * A)))
	tmp = 0.0
	if (B <= -0.49)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(t_2)) * t_4) / t_2);
	elseif (B <= -2.1e-88)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_5 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(Float64(B * B) / C)))))))) / t_5);
	elseif (B <= -2.15e-177)
		tmp = Float64(Float64(((Float64(t_2 * F) ^ 0.25) ^ 2.0) * t_4) / t_2);
	elseif (B <= -3.4e-270)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_5 * Float64(F * Float64(A * 2.0)))))) / t_5);
	elseif (B <= 8e-293)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(2.0 * Float64(C * F))))) * Float64(-1.0 / t_5));
	elseif (B <= 4.8e-268)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F))))))) / t_0);
	elseif (B <= 4.9e-110)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_5 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(B / Float64(C / B))))))))) / t_5);
	elseif (B <= 1.25e+74)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(-4.0 * Float64(A * C)))))))) / t_0);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = (-N[Sqrt[N[(2.0 * N[(t$95$1 + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$5 = N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -0.49], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, -2.1e-88], N[((-N[Sqrt[N[(2.0 * N[(t$95$5 * N[(F * N[(2.0 * A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[B, -2.15e-177], N[(N[(N[Power[N[Power[N[(t$95$2 * F), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, -3.4e-270], N[((-N[Sqrt[N[(2.0 * N[(t$95$5 * N[(F * N[(A * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[B, 8e-293], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.8e-268], N[(N[(t$95$3 * (-N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 4.9e-110], N[((-N[Sqrt[N[(2.0 * N[(t$95$5 * N[(F * N[(2.0 * A + N[(-0.5 * N[(B / N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[B, 1.25e+74], N[(N[(t$95$3 * (-N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if B < -0.48999999999999999

   1. Initial program 54.8

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

   2. Simplified 52.7

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

      [Start] 54.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 48.5

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

   4. Applied egg-rr 45.5

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

   if -0.48999999999999999 < B < -2.1e-88

   1. Initial program 43.3

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

   2. Simplified 37.5

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

      [Start] 43.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 49.6

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

   4. Simplified 49.6

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

      [Start] 49.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 49.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 49.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -2.1e-88 < B < -2.1500000000000001e-177

   1. Initial program 52.2

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

   2. Simplified 45.7

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

      [Start] 52.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 41.9

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

   4. Applied egg-rr 42.0

      \[\leadsto \frac{-\color{blue}{{\left({\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}^{0.25}\right)}^{2}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if -2.1500000000000001e-177 < B < -3.4000000000000001e-270

   1. Initial program 54.0

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

   2. Simplified 47.4

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

      [Start] 54.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 48.7

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

   if -3.4000000000000001e-270 < B < 8.0000000000000004e-293

   1. Initial program 54.9

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

   2. Simplified 48.6

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

      [Start] 54.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 49.7

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

   4. Applied egg-rr 49.9

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

   if 8.0000000000000004e-293 < B < 4.7999999999999998e-268

   1. Initial program 55.1

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

   2. Simplified 55.1

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

      [Start] 55.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 43.2

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

   4. Simplified 43.2

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

      [Start] 43.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 43.2	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 43.2	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around 0 45.3

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

   if 4.7999999999999998e-268 < B < 4.8999999999999997e-110

   1. Initial program 52.0

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

   2. Simplified 47.1

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

      [Start] 52.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 48.9

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

   4. Simplified 48.9

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

      [Start] 48.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 48.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 48.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-/l* [=>] 48.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \color{blue}{\frac{B}{\frac{C}{B}}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 4.8999999999999997e-110 < B < 1.24999999999999991e74

   1. Initial program 43.1

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

   2. Simplified 43.1

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

      [Start] 43.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 33.1

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

   4. Simplified 33.1

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

      [Start] 33.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 33.1	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 33.1	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.24999999999999991e74 < B

   1. Initial program 59.2

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

   2. Simplified 59.2

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

      [Start] 59.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 55.4

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

   4. Simplified 55.4

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

      [Start] 55.4	\[ \frac{-\sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 55.4	\[ \frac{-\sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 55.4	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 55.4	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 55.4	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)} \cdot \color{blue}{\left(B \cdot \sqrt{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 33.2

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

   6. Simplified 33.2

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

      [Start] 33.2	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      mul-1-neg [=>] 33.2	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      distribute-rgt-neg-in [=>] 33.2	\[ \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]

3. Recombined 9 regimes into one program.

4. Final simplification 42.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.49:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right) \cdot \left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}^{0.25}\right)}^{2} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-268}:\\ \;\;\;\;\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)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	42.2
Cost	40388

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{2 \cdot \left(t_2 + \left(A + C\right)\right)}\\ t_4 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_5 := \sqrt{A + \left(C + t_2\right)}\\ \mathbf{if}\;B \leq -0.022:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{t_4}\right) \cdot \left(-t_3\right)}{t_4}\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.52 \cdot 10^{-176}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{t_4 \cdot F}\right)}{t_4}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-265}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 2
Error	42.4
Cost	34252

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_2 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;A \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;A \leq -8 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(t_0 + \left(A + C\right)\right)} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_3}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-t_2\right)}{t_1}\\ \end{array} \]

Alternative 3
Error	43.3
Cost	33988

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_3 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_4 := \sqrt{t_3}\\ \mathbf{if}\;B \leq -1.28 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot \left(F \cdot 2\right)} \cdot \left(-\sqrt{t_2}\right)}{t_2}\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-275}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-269}:\\ \;\;\;\;\frac{t_4 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{t_4 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 4
Error	44.0
Cost	28312

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_0}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \sqrt{A + \left(C + t_3\right)}\\ \mathbf{if}\;B \leq -4.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{t_4 \cdot \left(\sqrt{2} \cdot t_1\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-188}:\\ \;\;\;\;\frac{t_4 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-277}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(C + \left(A + t_3\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 5
Error	43.9
Cost	28312

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_0}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \sqrt{A + \left(C + t_3\right)}\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{t_4 \cdot \left(\sqrt{2} \cdot t_1\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-188}:\\ \;\;\;\;\frac{t_4 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-277}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 10^{-285}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(C + \left(A + t_3\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 6
Error	43.2
Cost	28116

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \left(\left(B \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot t_1}\right)\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-275}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot t_1\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	40.2
Cost	28112

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \frac{t_1 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{if}\;B \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\frac{t_1 \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.28 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 8
Error	42.5
Cost	27732

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_3\right)} \cdot \left(\sqrt{2} \cdot t_1\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(C + \left(A + t_3\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9
Error	42.6
Cost	27732

\[\begin{array}{l} t_0 := B \cdot \sqrt{F}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{t_3} \cdot \left(t_0 \cdot \sqrt{2 \cdot \left(A + \left(C + t_2\right)\right)}\right)\\ \mathbf{elif}\;B \leq -3.2 \cdot 10^{-273}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-45}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(C + \left(A + t_2\right)\right)}}{\frac{t_3}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10
Error	44.4
Cost	26884

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ t_2 := \left(A \cdot C\right) \cdot 4\\ t_3 := B \cdot B - t_2\\ \mathbf{if}\;B \leq -2.6 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{B \cdot \sqrt{2}}{t_0}\\ \mathbf{elif}\;B \leq -6.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_3}\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_2 - B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11
Error	43.3
Cost	26884

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-274}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{B \cdot B - t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12
Error	44.6
Cost	21924

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \left(A \cdot C\right) \cdot 4\\ t_3 := B \cdot B - t_2\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ t_6 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_2 - B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{if}\;B \leq -6.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{-B} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_3}\\ \mathbf{elif}\;B \leq -700000000000:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-165}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-189}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.9 \cdot 10^{-253}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-261}:\\ \;\;\;\;\frac{-{\left(\sqrt{A} \cdot {\left(C \cdot \left(F \cdot -16\right)\right)}^{0.25}\right)}^{2}}{t_3}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-260}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+45}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13
Error	44.6
Cost	21660

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ t_4 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{-B} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_2}\\ \mathbf{elif}\;B \leq -750000000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-136}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -9.6 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-271}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{t_0 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-143}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+46}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 14
Error	44.7
Cost	21528

\[\begin{array}{l} t_0 := \left(A \cdot C\right) \cdot 4\\ t_1 := B \cdot B - t_0\\ t_2 := \frac{\sqrt{-B} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_1}\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ t_4 := C \cdot \left(-4 \cdot A\right)\\ t_5 := \mathsf{fma}\left(B, B, t_4\right)\\ t_6 := \frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_5}\\ \mathbf{if}\;B \leq -8.8 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-278}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-286}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-143}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 15
Error	45.1
Cost	21528

\[\begin{array}{l} t_0 := \left(A \cdot C\right) \cdot 4\\ t_1 := B \cdot B - t_0\\ t_2 := \sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_4}\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{-B} \cdot t_2}{t_1}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{A + \left(A + \frac{B \cdot \left(B \cdot -0.5\right)}{C}\right)} \cdot t_2}{t_1}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-277}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-290}:\\ \;\;\;\;\sqrt{t_4 \cdot \left(2 \cdot \left(2 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{t_4}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-145}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 16
Error	43.5
Cost	20484

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{-B} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_2}\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-279}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+46}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 17
Error	43.9
Cost	20292

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{B \cdot B - t_1}\\ \mathbf{if}\;B \leq -8.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-163}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 18
Error	43.9
Cost	20292

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{B \cdot B - t_1}\\ \mathbf{if}\;B \leq -4.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 19
Error	44.3
Cost	15308

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ t_1 := \left(A \cdot C\right) \cdot 4\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{B \cdot B - t_1}\\ \mathbf{if}\;B \leq -3.3 \cdot 10^{-282}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-165}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 20
Error	49.5
Cost	14604

\[\begin{array}{l} t_0 := C \cdot \left(-4 \cdot A\right)\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 21
Error	50.5
Cost	14476

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8.1 \cdot 10^{-293}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \left(F \cdot -8\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 22
Error	50.6
Cost	14348

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-284}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 23
Error	50.5
Cost	13448

\[\begin{array}{l} \mathbf{if}\;B \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-54}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 24
Error	51.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 25
Error	52.9
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 4.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) + \left(C - A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 26
Error	53.1
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 27
Error	55.7
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;C \leq -6.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 28
Error	56.5
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;B \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 29
Error	56.9
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;C \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C - A \cdot -2\right)\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 30
Error	56.6
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;C \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 31
Error	58.2
Cost	7808

\[\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4} \]

Alternative 32
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023039 
(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))))