ABCF->ab-angle b

Average Error: 52.3 → 41.8

Time: 1.1min

Precision: binary64

Cost: 60036

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(-2 \cdot F\right)\\ t_1 := e^{0.25 \cdot \left(t_0 - \log \left(\frac{-1}{B}\right)\right)}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ t_3 := \mathsf{hypot}\left(B, \sqrt{t_2}\right)\\ t_4 := \mathsf{fma}\left(B, B, t_2\right)\\ t_5 := \frac{-\sqrt{t_4 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{t_4}\\ t_6 := \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\frac{t_4}{\sqrt{2 \cdot t_4}}}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+176}:\\ \;\;\;\;-\frac{{\left(t_1 + 0.25 \cdot \frac{\left(A + C\right) \cdot t_1}{B}\right)}^{2}}{t_3}\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-194}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-308}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 5.9 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_4}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-184}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+150}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+228}:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(t_0 - \log \left(\frac{1}{B}\right)\right)}\right)}^{2}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (log (* -2.0 F)))
        (t_1 (exp (* 0.25 (- t_0 (log (/ -1.0 B))))))
        (t_2 (* A (* C -4.0)))
        (t_3 (hypot B (sqrt t_2)))
        (t_4 (fma B B t_2))
        (t_5 (/ (- (sqrt (* t_4 (* (* A 2.0) (* F 2.0))))) t_4))
        (t_6
         (/
          (- (sqrt (* F (+ C (- A (hypot B (- A C)))))))
          (/ t_4 (sqrt (* 2.0 t_4))))))
   (if (<= B -3.4e+176)
     (- (/ (pow (+ t_1 (* 0.25 (/ (* (+ A C) t_1) B))) 2.0) t_3))
     (if (<= B -1.02e-194)
       t_6
       (if (<= B 6.5e-308)
         t_5
         (if (<= B 5.9e-241)
           (/ (* (sqrt (* 2.0 (* F (* A C)))) (- (sqrt (* A -8.0)))) t_4)
           (if (<= B 2.5e-228)
             (/ (- (sqrt (* 2.0 (* -4.0 (* (* A C) (* F (+ C C))))))) t_4)
             (if (<= B 1.1e-184)
               t_5
               (if (<= B 4e-111)
                 (/ (- (sqrt (* 2.0 (* F (- C (hypot C B)))))) t_3)
                 (if (<= B 4.1e-44)
                   (/
                    (- (sqrt (* 2.0 (* -8.0 (* F (* A (* C C)))))))
                    (fma B B (* C (* A -4.0))))
                   (if (<= B 3.9e+150)
                     t_6
                     (if (<= B 3.4e+228)
                       (/
                        (- (pow (exp (* 0.25 (- t_0 (log (/ 1.0 B))))) 2.0))
                        t_3)
                       (*
                        (/ (sqrt 2.0) B)
                        (- (sqrt (* F (- A (hypot A 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 = log((-2.0 * F));
	double t_1 = exp((0.25 * (t_0 - log((-1.0 / B)))));
	double t_2 = A * (C * -4.0);
	double t_3 = hypot(B, sqrt(t_2));
	double t_4 = fma(B, B, t_2);
	double t_5 = -sqrt((t_4 * ((A * 2.0) * (F * 2.0)))) / t_4;
	double t_6 = -sqrt((F * (C + (A - hypot(B, (A - C)))))) / (t_4 / sqrt((2.0 * t_4)));
	double tmp;
	if (B <= -3.4e+176) {
		tmp = -(pow((t_1 + (0.25 * (((A + C) * t_1) / B))), 2.0) / t_3);
	} else if (B <= -1.02e-194) {
		tmp = t_6;
	} else if (B <= 6.5e-308) {
		tmp = t_5;
	} else if (B <= 5.9e-241) {
		tmp = (sqrt((2.0 * (F * (A * C)))) * -sqrt((A * -8.0))) / t_4;
	} else if (B <= 2.5e-228) {
		tmp = -sqrt((2.0 * (-4.0 * ((A * C) * (F * (C + C)))))) / t_4;
	} else if (B <= 1.1e-184) {
		tmp = t_5;
	} else if (B <= 4e-111) {
		tmp = -sqrt((2.0 * (F * (C - hypot(C, B))))) / t_3;
	} else if (B <= 4.1e-44) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / fma(B, B, (C * (A * -4.0)));
	} else if (B <= 3.9e+150) {
		tmp = t_6;
	} else if (B <= 3.4e+228) {
		tmp = -pow(exp((0.25 * (t_0 - log((1.0 / B))))), 2.0) / t_3;
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A - hypot(A, 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 = log(Float64(-2.0 * F))
	t_1 = exp(Float64(0.25 * Float64(t_0 - log(Float64(-1.0 / B)))))
	t_2 = Float64(A * Float64(C * -4.0))
	t_3 = hypot(B, sqrt(t_2))
	t_4 = fma(B, B, t_2)
	t_5 = Float64(Float64(-sqrt(Float64(t_4 * Float64(Float64(A * 2.0) * Float64(F * 2.0))))) / t_4)
	t_6 = Float64(Float64(-sqrt(Float64(F * Float64(C + Float64(A - hypot(B, Float64(A - C))))))) / Float64(t_4 / sqrt(Float64(2.0 * t_4))))
	tmp = 0.0
	if (B <= -3.4e+176)
		tmp = Float64(-Float64((Float64(t_1 + Float64(0.25 * Float64(Float64(Float64(A + C) * t_1) / B))) ^ 2.0) / t_3));
	elseif (B <= -1.02e-194)
		tmp = t_6;
	elseif (B <= 6.5e-308)
		tmp = t_5;
	elseif (B <= 5.9e-241)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(A * C)))) * Float64(-sqrt(Float64(A * -8.0)))) / t_4);
	elseif (B <= 2.5e-228)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-4.0 * Float64(Float64(A * C) * Float64(F * Float64(C + C))))))) / t_4);
	elseif (B <= 1.1e-184)
		tmp = t_5;
	elseif (B <= 4e-111)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B)))))) / t_3);
	elseif (B <= 4.1e-44)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(A * Float64(C * C))))))) / fma(B, B, Float64(C * Float64(A * -4.0))));
	elseif (B <= 3.9e+150)
		tmp = t_6;
	elseif (B <= 3.4e+228)
		tmp = Float64(Float64(-(exp(Float64(0.25 * Float64(t_0 - log(Float64(1.0 / B))))) ^ 2.0)) / t_3);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A - hypot(A, 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[Log[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(0.25 * N[(t$95$0 - N[Log[N[(-1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[B ^ 2 + N[Sqrt[t$95$2], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[(B * B + t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(t$95$4 * N[(N[(A * 2.0), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[((-N[Sqrt[N[(F * N[(C + N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$4 / N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.4e+176], (-N[(N[Power[N[(t$95$1 + N[(0.25 * N[(N[(N[(A + C), $MachinePrecision] * t$95$1), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), If[LessEqual[B, -1.02e-194], t$95$6, If[LessEqual[B, 6.5e-308], t$95$5, If[LessEqual[B, 5.9e-241], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A * -8.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 2.5e-228], N[((-N[Sqrt[N[(2.0 * N[(-4.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 1.1e-184], t$95$5, If[LessEqual[B, 4e-111], N[((-N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 4.1e-44], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.9e+150], t$95$6, If[LessEqual[B, 3.4e+228], N[((-N[Power[N[Exp[N[(0.25 * N[(t$95$0 - N[Log[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if B < -3.40000000000000014e176

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 63.4

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

   4. Simplified 63.4

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

      [Start] 63.4	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 63.4	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 46.4

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

   6. Simplified 46.2

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

      [Start] 46.4	\[ \frac{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-inverses [=>] 46.2	\[ \color{blue}{1} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-lft-identity [=>] 46.2	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}} \]
      associate-*r* [=>] 46.2	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot 2\right) \cdot F}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-commutative [=>] 46.2	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 46.2	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot F\right)}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate--l+ [=>] 46.2	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 46.2	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{A \cdot \left(-4 \cdot C\right)}}\right)} \]
      *-commutative [<=] 46.2	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \color{blue}{\left(C \cdot -4\right)}}\right)} \]

   7. Applied egg-rr 46.3

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

   8. Taylor expanded in B around -inf 40.1

      \[\leadsto \frac{-{\color{blue}{\left(0.25 \cdot \frac{\left(A + C\right) \cdot e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) + -1 \cdot \log \left(\frac{-1}{B}\right)\right)}}{B} + e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) + -1 \cdot \log \left(\frac{-1}{B}\right)\right)}\right)}}^{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]

   if -3.40000000000000014e176 < B < -1.02e-194 or 4.09999999999999992e-44 < B < 3.89999999999999991e150

   1. Initial program 46.9

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

   2. Simplified 42.9

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

      [Start] 46.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 38.5

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

   4. Simplified 38.2

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

      [Start] 38.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 38.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate--l+ [=>] 38.2	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 38.6

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

   6. Simplified 38.2

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

      [Start] 38.6	\[ \left(-\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}\right) \]
      associate-*r/ [=>] 38.5	\[ \left(-\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)} \cdot 1}{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}} \]
      *-rgt-identity [=>] 38.5	\[ \left(-\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}}}{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)} \]
      associate-*r/ [=>] 38.5	\[ \color{blue}{\frac{\left(-\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}}{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}} \]
      associate-/l* [=>] 38.5	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}}}} \]
      associate--l+ [=>] 38.2	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\frac{\mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}}} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \left(C \cdot -4\right) \cdot A\right)}}} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)}}} \]

   if -1.02e-194 < B < 6.4999999999999999e-308 or 2.49999999999999986e-228 < B < 1.09999999999999996e-184

   1. Initial program 52.5

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

   2. Simplified 46.0

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

      [Start] 52.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 48.4

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

   if 6.4999999999999999e-308 < B < 5.8999999999999998e-241

   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 47.3

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      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 51.1

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

   4. Simplified 51.1

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

      [Start] 51.1	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 51.1	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 51.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 49.1

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

   6. Simplified 49.1

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

      [Start] 49.1	\[ \frac{-\sqrt{-8 \cdot A} \cdot \sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)} \cdot \sqrt{-8 \cdot A}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 50.6	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot C\right) \cdot \left(A + A\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      distribute-rgt-out [<=] 50.6	\[ \frac{-\sqrt{\color{blue}{A \cdot \left(F \cdot C\right) + A \cdot \left(F \cdot C\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      count-2 [=>] 50.6	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(A \cdot \left(F \cdot C\right)\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 50.6	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(F \cdot C\right) \cdot A\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(C \cdot A\right)\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [<=] 49.1	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(A \cdot C\right)}\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{A \cdot -8}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 5.8999999999999998e-241 < B < 2.49999999999999986e-228

   1. Initial program 56.9

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

   2. Simplified 49.7

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

      [Start] 56.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 49.4

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

   4. Simplified 45.5

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

      [Start] 49.4	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 45.5	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 45.5	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \color{blue}{\left(F \cdot \left(C - -1 \cdot C\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 45.5	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 1.09999999999999996e-184 < B < 4.00000000000000035e-111

   1. Initial program 51.4

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

   2. Simplified 46.2

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

      [Start] 51.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.6

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

   4. Simplified 48.8

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

      [Start] 49.6	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 48.8	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 48.7

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

   6. Simplified 47.8

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

      [Start] 48.7	\[ \frac{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-inverses [=>] 48.6	\[ \color{blue}{1} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-lft-identity [=>] 48.6	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}} \]
      associate-*r* [=>] 48.6	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot 2\right) \cdot F}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-commutative [=>] 48.6	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 48.6	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot F\right)}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate--l+ [=>] 47.8	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 47.8	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{A \cdot \left(-4 \cdot C\right)}}\right)} \]
      *-commutative [<=] 47.8	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \color{blue}{\left(C \cdot -4\right)}}\right)} \]

   7. Applied egg-rr 47.9

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

   8. Taylor expanded in A around 0 52.7

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

   9. Simplified 50.6

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

      [Start] 52.7	\[ \frac{-\sqrt{2 \cdot \left(\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      +-commutative [=>] 52.7	\[ \frac{-\sqrt{2 \cdot \left(\left(C - \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      unpow2 [=>] 52.7	\[ \frac{-\sqrt{2 \cdot \left(\left(C - \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      unpow2 [=>] 52.7	\[ \frac{-\sqrt{2 \cdot \left(\left(C - \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      hypot-def [=>] 50.6	\[ \frac{-\sqrt{2 \cdot \left(\left(C - \color{blue}{\mathsf{hypot}\left(C, B\right)}\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]

   if 4.00000000000000035e-111 < B < 4.09999999999999992e-44

   1. Initial program 46.4

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

   2. Simplified 40.9

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

      [Start] 46.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 56.6

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

   4. Simplified 55.8

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

      [Start] 56.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 55.8	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 55.8	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 3.89999999999999991e150 < B < 3.3999999999999998e228

   1. Initial program 63.7

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

   2. Simplified 63.4

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

      [Start] 63.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 62.4

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

   4. Simplified 62.4

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

      [Start] 62.4	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 62.4	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 43.8

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

   6. Simplified 43.7

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

      [Start] 43.8	\[ \frac{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-inverses [=>] 43.7	\[ \color{blue}{1} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-lft-identity [=>] 43.7	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)}} \]
      associate-*r* [=>] 43.7	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot 2\right) \cdot F}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      *-commutative [=>] 43.7	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 43.7	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot F\right)}}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate--l+ [=>] 43.7	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 43.7	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{A \cdot \left(-4 \cdot C\right)}}\right)} \]
      *-commutative [<=] 43.7	\[ \frac{-\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \color{blue}{\left(C \cdot -4\right)}}\right)} \]

   7. Applied egg-rr 43.8

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

   8. Taylor expanded in B around inf 39.8

      \[\leadsto \frac{-{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)}\right)}}^{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]

   if 3.3999999999999998e228 < B

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(A - \sqrt{B \cdot B + A \cdot A}\right) \cdot \left(F \cdot \color{blue}{\left(B \cdot B\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Taylor expanded in C around 0 64.0

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

   6. Simplified 31.7

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

      [Start] 64.0	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 64.0	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]
      +-commutative [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{{A}^{2} + {B}^{2}}}\right)}\right) \]
      unpow2 [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]
      unpow2 [=>] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right)}\right) \]
      hypot-def [=>] 31.7	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right)}\right) \]

3. Recombined 9 regimes into one program.

4. Final simplification 41.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{+176}:\\ \;\;\;\;-\frac{{\left(e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) - \log \left(\frac{-1}{B}\right)\right)} + 0.25 \cdot \frac{\left(A + C\right) \cdot e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) - \log \left(\frac{-1}{B}\right)\right)}}{B}\right)}^{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-194}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.9 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+228}:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(\log \left(-2 \cdot F\right) - \log \left(\frac{1}{B}\right)\right)}\right)}^{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	41.2
Cost	40740

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \log \left(-2 \cdot F\right)\\ t_4 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ t_6 := \frac{-\sqrt{F \cdot \left(C + \left(A - t_5\right)\right)}}{\frac{t_2}{\sqrt{2 \cdot t_2}}}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+169}:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(t_3 - \log \left(\frac{-1}{B}\right)\right)}\right)}^{2}}{t_4}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-194}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{\left(F \cdot \left(A \cdot -4\right)\right) \cdot \left(C \cdot \left(A + A\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(t_5 - C\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot -16\right) \cdot \left(A \cdot A\right)\right)}}{-t_0}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+228}:\\ \;\;\;\;\frac{-{\left(e^{0.25 \cdot \left(t_3 - \log \left(\frac{1}{B}\right)\right)}\right)}^{2}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	42.5
Cost	34120

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;A \leq -6.4 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{\left(A + \left(C - t_3\right)\right) \cdot \left(F \cdot t_2\right)}\right)}{t_2}\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-126}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\left(t_3 - C\right) - A\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot -4} \cdot \sqrt{A}\right)}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)\right)}}{t_2}\\ \end{array} \]

Alternative 3
Error	43.0
Cost	33808

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ \mathbf{if}\;A \leq -2.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -2.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(\left(C + \left(A - t_3\right)\right) \cdot t_0\right)}}{-t_0}\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\left(t_3 - C\right) - A\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot -4} \cdot \sqrt{A}\right)}\\ \mathbf{elif}\;A \leq 4.7 \cdot 10^{+142}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)\right)}}{t_2}\\ \end{array} \]

Alternative 4
Error	42.2
Cost	28308

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ \mathbf{if}\;A \leq -3.9 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(\left(C + \left(A - t_3\right)\right) \cdot t_0\right)}}{-t_0}\\ \mathbf{elif}\;A \leq -2.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 10^{-125}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\left(t_3 - C\right) - A\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 1.06 \cdot 10^{+142}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)\right)}}{t_2}\\ \end{array} \]

Alternative 5
Error	43.6
Cost	27804

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;B \leq -4.7 \cdot 10^{+86}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\frac{-1}{\frac{1}{C - t_2}} - A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -2.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_3 \cdot \left(F \cdot \left(\left(t_2 - A\right) - C\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_3}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-155}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(F \cdot 2\right) \cdot \left(C \cdot 2\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.56 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\left(t_2 - C\right) - A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 6
Error	42.4
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ \mathbf{if}\;A \leq -9.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 5.3 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 7
Error	42.7
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ \mathbf{if}\;A \leq -2.7 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -3.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(\left(C + \left(A - t_3\right)\right) \cdot t_0\right)}}{-t_0}\\ \mathbf{elif}\;A \leq -2.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 10^{-125}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\left(t_3 - C\right) - A\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 6 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 8
Error	43.1
Cost	27020

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ \mathbf{if}\;A \leq -8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_2}\\ \mathbf{elif}\;A \leq -1.72 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 5.9 \cdot 10^{+141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 9
Error	44.3
Cost	26756

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ t_3 := \mathsf{fma}\left(B, B, t_1\right)\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+101}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{t_2}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-293}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(F \cdot 2\right) \cdot \left(C \cdot 2\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 10
Error	44.5
Cost	21832

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ t_3 := \mathsf{fma}\left(B, B, t_1\right)\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-193}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{t_2}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-295}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot 2\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(F \cdot 2\right) \cdot \left(C \cdot 2\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 11
Error	44.4
Cost	21264

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-69}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(F \cdot 2\right) \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 12
Error	44.5
Cost	20876

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -9 \cdot 10^{+100}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-69}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 13
Error	44.7
Cost	20432

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;B \leq -8.6 \cdot 10^{-59}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 14
Error	44.7
Cost	20424

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-125}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 15
Error	48.7
Cost	20300

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{C \cdot C + B \cdot B} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -7 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}\right)\\ \mathbf{elif}\;A \leq 8.8 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 16
Error	45.0
Cost	20300

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -7 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{t_1}\right)}\\ \mathbf{elif}\;B \leq -2.25 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{C \cdot C + B \cdot B} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 17
Error	47.2
Cost	20168

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{C \cdot C + B \cdot B} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)\\ \end{array} \]

Alternative 18
Error	48.7
Cost	15500

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -1.65 \cdot 10^{+95}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 2.1 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C - \sqrt{A \cdot A + B \cdot B}\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 19
Error	48.6
Cost	15368

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{C \cdot C + B \cdot B} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 20
Error	51.3
Cost	14476

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;C \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 21
Error	53.0
Cost	14348

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_2}\\ \mathbf{elif}\;C \leq -3.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \end{array} \]

Alternative 22
Error	52.2
Cost	14348

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;C \leq -3.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 4.3 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \end{array} \]

Alternative 23
Error	52.0
Cost	14348

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;C \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;C \leq -3.7 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-t_2}\\ \end{array} \]

Alternative 24
Error	53.7
Cost	14216

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + \left(C - A\right)\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 25
Error	55.6
Cost	9224

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -3.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + \left(\left(C - A\right) - -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 26
Error	55.6
Cost	8712

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + \left(C - A\right)\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(B - C\right) - A\right)\right)}}{t_1}\\ \end{array} \]

Alternative 27
Error	55.1
Cost	8584

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(B - C\right) - A\right)\right)}}{t_1}\\ \end{array} \]

Alternative 28
Error	55.1
Cost	8584

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(B - C\right) - A\right)\right)}}{t_1}\\ \end{array} \]

Alternative 29
Error	56.6
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 30
Error	59.3
Cost	8320

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

Alternative 31
Error	61.8
Cost	7808

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

Alternative 32
Error	61.6
Cost	6464

\[\sqrt{0} \]

Reproduce

herbie shell --seed 2023066 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))