ABCF->ab-angle a

Average Error: 81.25% → 60.59%

Time: 52.0s

Precision: binary64

Cost: 40716

Math

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

↓

\[\begin{array}{l} t_0 := \left(B \cdot B\right) \cdot -0.5\\ t_1 := \left(C \cdot A\right) \cdot -4\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_5 := B \cdot B + t_1\\ t_6 := \mathsf{hypot}\left(B, A - C\right)\\ t_7 := A + \left(C + t_6\right)\\ t_8 := \sqrt{2 \cdot \left(F \cdot t_2\right)}\\ \mathbf{if}\;B \leq -3.75 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{t_7} \cdot \left(-t_8\right)}{t_5}\\ \mathbf{elif}\;B \leq 3.45 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, A, \frac{t_0}{C}\right)} \cdot \left(\sqrt{t_4} \cdot \left(-\sqrt{F}\right)\right)}{t_4}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A + t_6\right)\right)} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_3}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-79}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B}{\frac{C}{B}}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{t_8 \cdot \left(-\sqrt{2 \cdot A}\right)}{t_5}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot t_7\right) \cdot t_2\right)} \cdot \frac{-1}{t_2}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+99}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{t_0}{A}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{fma}\left(0.5, \frac{C \cdot C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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) -0.5))
        (t_1 (* (* C A) -4.0))
        (t_2 (fma B B t_1))
        (t_3 (fma B B (* C (* A -4.0))))
        (t_4 (fma -4.0 (* C A) (* B B)))
        (t_5 (+ (* B B) t_1))
        (t_6 (hypot B (- A C)))
        (t_7 (+ A (+ C t_6)))
        (t_8 (sqrt (* 2.0 (* F t_2)))))
   (if (<= B -3.75e+124)
     (/ (sqrt (* F (+ C (hypot B C)))) (/ B (sqrt 2.0)))
     (if (<= B 1.45e-159)
       (/ (* (sqrt t_7) (- t_8)) t_5)
       (if (<= B 3.45e-118)
         (/
          (* (sqrt (* 2.0 (fma 2.0 A (/ t_0 C)))) (* (sqrt t_4) (- (sqrt F))))
          t_4)
         (if (<= B 5e-99)
           (/ (* (sqrt (* F (+ C (+ A t_6)))) (- (sqrt (* 2.0 t_3)))) t_3)
           (if (<= B 1.4e-79)
             (-
              (/
               (sqrt (* 2.0 (* t_3 (* F (fma 2.0 A (* -0.5 (/ B (/ C B))))))))
               t_3))
             (if (<= B 5.8e-78)
               (/ (* t_8 (- (sqrt (* 2.0 A)))) t_5)
               (if (<= B 2.8e+32)
                 (* (sqrt (* 2.0 (* (* F t_7) t_2))) (/ -1.0 t_2))
                 (if (<= B 1.95e+99)
                   (/
                    (- (sqrt (* 2.0 (* t_3 (* F (fma 2.0 C (/ t_0 A)))))))
                    t_3)
                   (if (<= B 1.45e+199)
                     (* (sqrt 2.0) (- (sqrt (/ F B))))
                     (*
                      (sqrt (* F (+ C (fma 0.5 (/ (* C C) B) B))))
                      (/ (- (sqrt 2.0)) 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) * -0.5;
	double t_1 = (C * A) * -4.0;
	double t_2 = fma(B, B, t_1);
	double t_3 = fma(B, B, (C * (A * -4.0)));
	double t_4 = fma(-4.0, (C * A), (B * B));
	double t_5 = (B * B) + t_1;
	double t_6 = hypot(B, (A - C));
	double t_7 = A + (C + t_6);
	double t_8 = sqrt((2.0 * (F * t_2)));
	double tmp;
	if (B <= -3.75e+124) {
		tmp = sqrt((F * (C + hypot(B, C)))) / (B / sqrt(2.0));
	} else if (B <= 1.45e-159) {
		tmp = (sqrt(t_7) * -t_8) / t_5;
	} else if (B <= 3.45e-118) {
		tmp = (sqrt((2.0 * fma(2.0, A, (t_0 / C)))) * (sqrt(t_4) * -sqrt(F))) / t_4;
	} else if (B <= 5e-99) {
		tmp = (sqrt((F * (C + (A + t_6)))) * -sqrt((2.0 * t_3))) / t_3;
	} else if (B <= 1.4e-79) {
		tmp = -(sqrt((2.0 * (t_3 * (F * fma(2.0, A, (-0.5 * (B / (C / B)))))))) / t_3);
	} else if (B <= 5.8e-78) {
		tmp = (t_8 * -sqrt((2.0 * A))) / t_5;
	} else if (B <= 2.8e+32) {
		tmp = sqrt((2.0 * ((F * t_7) * t_2))) * (-1.0 / t_2);
	} else if (B <= 1.95e+99) {
		tmp = -sqrt((2.0 * (t_3 * (F * fma(2.0, C, (t_0 / A)))))) / t_3;
	} else if (B <= 1.45e+199) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else {
		tmp = sqrt((F * (C + fma(0.5, ((C * C) / B), B)))) * (-sqrt(2.0) / 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) * -0.5)
	t_1 = Float64(Float64(C * A) * -4.0)
	t_2 = fma(B, B, t_1)
	t_3 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_4 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_5 = Float64(Float64(B * B) + t_1)
	t_6 = hypot(B, Float64(A - C))
	t_7 = Float64(A + Float64(C + t_6))
	t_8 = sqrt(Float64(2.0 * Float64(F * t_2)))
	tmp = 0.0
	if (B <= -3.75e+124)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(B, C)))) / Float64(B / sqrt(2.0)));
	elseif (B <= 1.45e-159)
		tmp = Float64(Float64(sqrt(t_7) * Float64(-t_8)) / t_5);
	elseif (B <= 3.45e-118)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(2.0, A, Float64(t_0 / C)))) * Float64(sqrt(t_4) * Float64(-sqrt(F)))) / t_4);
	elseif (B <= 5e-99)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + Float64(A + t_6)))) * Float64(-sqrt(Float64(2.0 * t_3)))) / t_3);
	elseif (B <= 1.4e-79)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_3 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(B / Float64(C / B)))))))) / t_3));
	elseif (B <= 5.8e-78)
		tmp = Float64(Float64(t_8 * Float64(-sqrt(Float64(2.0 * A)))) / t_5);
	elseif (B <= 2.8e+32)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_7) * t_2))) * Float64(-1.0 / t_2));
	elseif (B <= 1.95e+99)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(F * fma(2.0, C, Float64(t_0 / A))))))) / t_3);
	elseif (B <= 1.45e+199)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(C + fma(0.5, Float64(Float64(C * C) / B), B)))) * Float64(Float64(-sqrt(2.0)) / 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] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(B * B), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$7 = N[(A + N[(C + t$95$6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -3.75e+124], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.45e-159], N[(N[(N[Sqrt[t$95$7], $MachinePrecision] * (-t$95$8)), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[B, 3.45e-118], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * A + N[(t$95$0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 5e-99], N[(N[(N[Sqrt[N[(F * N[(C + N[(A + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[B, 1.4e-79], (-N[(N[Sqrt[N[(2.0 * N[(t$95$3 * N[(F * N[(2.0 * A + N[(-0.5 * N[(B / N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), If[LessEqual[B, 5.8e-78], N[(N[(t$95$8 * (-N[Sqrt[N[(2.0 * A), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[B, 2.8e+32], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$7), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.95e+99], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(F * N[(2.0 * C + N[(t$95$0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 1.45e+199], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(C + N[(0.5 * N[(N[(C * C), $MachinePrecision] / B), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 10 regimes

2. if B < -3.75000000000000019e124

   1. Initial program 97

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

   2. Simplified 97

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

      [Start] 97	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 99.73

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

   4. Simplified 99.73

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

      [Start] 99.73	\[ \frac{-\sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 99.73	\[ \frac{-\sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 99.73	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 99.73	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 99.73	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)} \cdot \color{blue}{\left(B \cdot \sqrt{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 89.21

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

   6. Taylor expanded in B around inf 48.79

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

   if -3.75000000000000019e124 < B < 1.44999999999999995e-159

   1. Initial program 76.46

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

   2. Simplified 76.47

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

      [Start] 76.46	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 60.57

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

   4. Simplified 60.57

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

      [Start] 60.57	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 60.57	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 60.57	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.44999999999999995e-159 < B < 3.4500000000000001e-118

   1. Initial program 81.57

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

   2. Simplified 70.62

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

      [Start] 81.57	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 62.87

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

   4. Simplified 61.21

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

      [Start] 62.87	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 62.87	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 62.87	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 62.87	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 62.87	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 61.21	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 68.73

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

   6. Taylor expanded in C around -inf 76.82

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

   7. Simplified 76.82

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

      [Start] 76.82	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      fma-def [=>] 76.82	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*r/ [=>] 76.82	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(2, A, \color{blue}{\frac{-0.5 \cdot {B}^{2}}{C}}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      unpow2 [=>] 76.82	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(2, A, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{C}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 3.4500000000000001e-118 < B < 4.99999999999999969e-99

   1. Initial program 76.88

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

   2. Simplified 73.47

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

      [Start] 76.88	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 74.69

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

   4. Simplified 74.69

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

      [Start] 74.69	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 74.69	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 4.99999999999999969e-99 < B < 1.40000000000000006e-79

   1. Initial program 75.66

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

   2. Simplified 65.66

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

      [Start] 75.66	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 78.96

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

   4. Simplified 78.96

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

      [Start] 78.96	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 78.96	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 78.96	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-/l* [=>] 78.96	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \color{blue}{\frac{B}{\frac{C}{B}}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.40000000000000006e-79 < B < 5.8000000000000001e-78

   1. Initial program 54.32

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

   2. Simplified 54.32

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

      [Start] 54.32	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 54.99

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

   4. Simplified 54.99

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

      [Start] 54.99	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 54.99	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 54.99	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around inf 98.17

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

   if 5.8000000000000001e-78 < B < 2.8e32

   1. Initial program 62.75

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

   2. Simplified 56.29

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

      [Start] 62.75	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.42

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

   4. Simplified 48.53

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

      [Start] 49.42	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 49.42	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 49.42	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 49.42	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 49.42	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 48.53	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 54.69

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

   6. Applied egg-rr 56.26

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

   7. Simplified 56.77

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

      [Start] 56.26	\[ \sqrt{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

   if 2.8e32 < B < 1.94999999999999997e99

   1. Initial program 68.79

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

   2. Simplified 61.46

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

      [Start] 68.79	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 89.02

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

   4. Simplified 89.02

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

      [Start] 89.02	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 89.02	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [=>] 89.02	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 89.02	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.94999999999999997e99 < B < 1.4499999999999999e199

   1. Initial program 90.12

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

   2. Simplified 90.12

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

      [Start] 90.12	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 79.08

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

   4. Simplified 79.08

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

      [Start] 79.08	\[ \frac{-\sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 79.08	\[ \frac{-\sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 79.08	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 79.08	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \cdot \left(\sqrt{2} \cdot B\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 79.08	\[ \frac{-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)} \cdot \color{blue}{\left(B \cdot \sqrt{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 53.77

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

   6. Simplified 53.77

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

      [Start] 53.77	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      mul-1-neg [=>] 53.77	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      distribute-rgt-neg-in [=>] 53.77	\[ \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]

   if 1.4499999999999999e199 < B

   1. Initial program 100

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

   2. Simplified 100

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

      [Start] 100	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around inf 100

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

   4. Taylor expanded in A around 0 61.25

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}\right)} \]

   5. Simplified 61.25

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

      [Start] 61.25	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}\right) \]
      associate-*r* [=>] 61.25	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}} \]
      *-commutative [=>] 61.25	\[ \color{blue}{\sqrt{F \cdot \left(C + \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]
      fma-def [=>] 61.25	\[ \sqrt{F \cdot \left(C + \color{blue}{\mathsf{fma}\left(0.5, \frac{{C}^{2}}{B}, B\right)}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      unpow2 [=>] 61.25	\[ \sqrt{F \cdot \left(C + \mathsf{fma}\left(0.5, \frac{\color{blue}{C \cdot C}}{B}, B\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      associate-*r/ [=>] 61.25	\[ \sqrt{F \cdot \left(C + \mathsf{fma}\left(0.5, \frac{C \cdot C}{B}, B\right)\right)} \cdot \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \]
      mul-1-neg [=>] 61.25	\[ \sqrt{F \cdot \left(C + \mathsf{fma}\left(0.5, \frac{C \cdot C}{B}, B\right)\right)} \cdot \frac{\color{blue}{-\sqrt{2}}}{B} \]

3. Recombined 10 regimes into one program.

4. Final simplification 60.59

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.75 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-159}:\\ \;\;\;\;\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, \left(C \cdot A\right) \cdot -4\right)\right)}\right)}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B \leq 3.45 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, A, \frac{\left(B \cdot B\right) \cdot -0.5}{C}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-79}:\\ \;\;\;\;-\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{B}{\frac{C}{B}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+99}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{fma}\left(0.5, \frac{C \cdot C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	55.53%
Cost	40652

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot -4\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 10^{-173}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\right)}{B \cdot B + t_0}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(\sqrt{t_2} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 2
Error	61.39%
Cost	28312

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot -4\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ t_3 := \sqrt{2 \cdot \left(\left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot t_2\right)} \cdot \frac{-1}{t_2}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{-148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_2\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{B \cdot B + t_0}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+72}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 3
Error	62.03%
Cost	28248

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)\right)\right)}}{t_0}\\ t_3 := \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_3\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{B \cdot B + t_3}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 4
Error	61.76%
Cost	28248

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \left(C \cdot A\right) \cdot -4\\ t_2 := F \cdot t_0\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.18 \cdot 10^{+126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + \left(C + t_3\right)\right)\right) \cdot t_2}}{t_0}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{t_2} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{-159}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{B \cdot B + t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+71}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(t_3 + \left(C + A\right)\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 5
Error	56.9%
Cost	28112

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot -4\\ t_1 := \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\right)}{B \cdot B + t_0}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 6
Error	62.18%
Cost	27864

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(C \cdot A\right) \cdot -4\\ t_2 := B \cdot B + t_1\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{t_2}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \left(-\sqrt{F}\right)\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 7
Error	62.14%
Cost	27864

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_2} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \left(-\sqrt{F}\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 8
Error	62.08%
Cost	21396

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(C \cdot A\right) \cdot -4\\ t_2 := B \cdot B + t_1\\ t_3 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{-148}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{t_2}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9
Error	61.83%
Cost	19972

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10
Error	68.6%
Cost	15044

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;B \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11
Error	74.18%
Cost	14216

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;B \leq -2.15 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(C + A\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12
Error	78.29%
Cost	13448

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.62 \cdot 10^{-132}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-208}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) + \left(\left(A - C\right) + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13
Error	83.16%
Cost	8844

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -5.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 10^{-120}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	82.72%
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -9.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	87.22%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;C \leq -2.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	84.67%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;C \leq 3.3 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(C + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	89.23%
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;C \leq -1.28 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	88.45%
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;C \leq -2.7 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	96.59%
Cost	7808

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

Alternative 20
Error	91.06%
Cost	7808

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

Alternative 21
Error	97.55%
Cost	6656

\[-\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023121 
(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))))