ABCF->ab-angle b

Average Error: 81.97% → 72.14%

Time: 1.0min

Precision: binary64

Cost: 40868

Math

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

↓

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ t_5 := \sqrt{F \cdot \left(\left(A + C\right) - t_4\right)}\\ t_6 := -\sqrt{2}\\ t_7 := \frac{t_6}{t_2}\\ \mathbf{if}\;A \leq -6.8 \cdot 10^{+176}:\\ \;\;\;\;t_5 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ \mathbf{elif}\;A \leq -5.5 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(t_4 - C\right) - A\right)\right)} \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;A \leq -4 \cdot 10^{-85}:\\ \;\;\;\;t_5 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot t_7\\ \mathbf{elif}\;A \leq -5.8 \cdot 10^{-167}:\\ \;\;\;\;t_5 \cdot \frac{t_6}{B}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - t_4\right)\right) \cdot \left(F \cdot 2\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;t_5 \cdot \left(\sqrt{2} \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;A \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_3}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;t_5 \cdot \frac{t_6}{\mathsf{hypot}\left(B, \sqrt{C \cdot -4} \cdot \sqrt{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_7 \cdot \sqrt{F \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)}\\ \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (fma B B t_0))
        (t_2 (hypot B (sqrt t_0)))
        (t_3 (fma B B (* C (* A -4.0))))
        (t_4 (hypot B (- A C)))
        (t_5 (sqrt (* F (- (+ A C) t_4))))
        (t_6 (- (sqrt 2.0)))
        (t_7 (/ t_6 t_2)))
   (if (<= A -6.8e+176)
     (* t_5 (- (sqrt (/ 2.0 (fma B B (* (* A C) -4.0))))))
     (if (<= A -5.5e-44)
       (* (sqrt (* -2.0 (* (* F t_1) (- (- t_4 C) A)))) (/ 1.0 (- t_1)))
       (if (<= A -4e-85)
         (* t_5 (/ (sqrt 2.0) B))
         (if (<= A -1.6e-140)
           (* (sqrt (* F (* A 2.0))) t_7)
           (if (<= A -5.8e-167)
             (* t_5 (/ t_6 B))
             (if (<= A -4.6e-303)
               (/ (- (sqrt (* t_1 (* (+ A (- C t_4)) (* F 2.0))))) t_1)
               (if (<= A 7.8e-177)
                 (* t_5 (* (sqrt 2.0) (/ -1.0 t_2)))
                 (if (<= A 5.3e-102)
                   (/
                    (sqrt
                     (* 2.0 (* (* F t_3) (+ A (fma -0.5 (/ (* B B) C) A)))))
                    (- t_3))
                   (if (<= A 1.7e+92)
                     (* t_5 (/ t_6 (hypot B (* (sqrt (* C -4.0)) (sqrt A)))))
                     (*
                      t_7
                      (sqrt
                       (*
                        F
                        (fma -0.5 (/ B (/ (- A C) B)) (* C 2.0))))))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = fma(B, B, t_0);
	double t_2 = hypot(B, sqrt(t_0));
	double t_3 = fma(B, B, (C * (A * -4.0)));
	double t_4 = hypot(B, (A - C));
	double t_5 = sqrt((F * ((A + C) - t_4)));
	double t_6 = -sqrt(2.0);
	double t_7 = t_6 / t_2;
	double tmp;
	if (A <= -6.8e+176) {
		tmp = t_5 * -sqrt((2.0 / fma(B, B, ((A * C) * -4.0))));
	} else if (A <= -5.5e-44) {
		tmp = sqrt((-2.0 * ((F * t_1) * ((t_4 - C) - A)))) * (1.0 / -t_1);
	} else if (A <= -4e-85) {
		tmp = t_5 * (sqrt(2.0) / B);
	} else if (A <= -1.6e-140) {
		tmp = sqrt((F * (A * 2.0))) * t_7;
	} else if (A <= -5.8e-167) {
		tmp = t_5 * (t_6 / B);
	} else if (A <= -4.6e-303) {
		tmp = -sqrt((t_1 * ((A + (C - t_4)) * (F * 2.0)))) / t_1;
	} else if (A <= 7.8e-177) {
		tmp = t_5 * (sqrt(2.0) * (-1.0 / t_2));
	} else if (A <= 5.3e-102) {
		tmp = sqrt((2.0 * ((F * t_3) * (A + fma(-0.5, ((B * B) / C), A))))) / -t_3;
	} else if (A <= 1.7e+92) {
		tmp = t_5 * (t_6 / hypot(B, (sqrt((C * -4.0)) * sqrt(A))));
	} else {
		tmp = t_7 * sqrt((F * fma(-0.5, (B / ((A - C) / B)), (C * 2.0))));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = fma(B, B, t_0)
	t_2 = hypot(B, sqrt(t_0))
	t_3 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_4 = hypot(B, Float64(A - C))
	t_5 = sqrt(Float64(F * Float64(Float64(A + C) - t_4)))
	t_6 = Float64(-sqrt(2.0))
	t_7 = Float64(t_6 / t_2)
	tmp = 0.0
	if (A <= -6.8e+176)
		tmp = Float64(t_5 * Float64(-sqrt(Float64(2.0 / fma(B, B, Float64(Float64(A * C) * -4.0))))));
	elseif (A <= -5.5e-44)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(Float64(F * t_1) * Float64(Float64(t_4 - C) - A)))) * Float64(1.0 / Float64(-t_1)));
	elseif (A <= -4e-85)
		tmp = Float64(t_5 * Float64(sqrt(2.0) / B));
	elseif (A <= -1.6e-140)
		tmp = Float64(sqrt(Float64(F * Float64(A * 2.0))) * t_7);
	elseif (A <= -5.8e-167)
		tmp = Float64(t_5 * Float64(t_6 / B));
	elseif (A <= -4.6e-303)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + Float64(C - t_4)) * Float64(F * 2.0))))) / t_1);
	elseif (A <= 7.8e-177)
		tmp = Float64(t_5 * Float64(sqrt(2.0) * Float64(-1.0 / t_2)));
	elseif (A <= 5.3e-102)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_3) * Float64(A + fma(-0.5, Float64(Float64(B * B) / C), A))))) / Float64(-t_3));
	elseif (A <= 1.7e+92)
		tmp = Float64(t_5 * Float64(t_6 / hypot(B, Float64(sqrt(Float64(C * -4.0)) * sqrt(A)))));
	else
		tmp = Float64(t_7 * sqrt(Float64(F * fma(-0.5, Float64(B / Float64(Float64(A - C) / B)), Float64(C * 2.0)))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$7 = N[(t$95$6 / t$95$2), $MachinePrecision]}, If[LessEqual[A, -6.8e+176], N[(t$95$5 * (-N[Sqrt[N[(2.0 / N[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[A, -5.5e-44], N[(N[Sqrt[N[(-2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(N[(t$95$4 - C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -4e-85], N[(t$95$5 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -1.6e-140], N[(N[Sqrt[N[(F * N[(A * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$7), $MachinePrecision], If[LessEqual[A, -5.8e-167], N[(t$95$5 * N[(t$95$6 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -4.6e-303], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + N[(C - t$95$4), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[A, 7.8e-177], N[(t$95$5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 5.3e-102], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$3), $MachinePrecision] * N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[A, 1.7e+92], N[(t$95$5 * N[(t$95$6 / N[Sqrt[B ^ 2 + N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$7 * N[Sqrt[N[(F * N[(-0.5 * N[(B / N[(N[(A - C), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 10 regimes

2. if A < -6.80000000000000028e176

   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 83.19

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

      [Start] 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. Applied egg-rr 82.01

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

   4. Simplified 82.01

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

      [Start] 82.01	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 82.01	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 82.01	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 82.03

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

   6. Simplified 82.02

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

      [Start] 82.03	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 82.03	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 82.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 82.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Applied egg-rr 80.3

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

   8. Simplified 79.65

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

      [Start] 80.3	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      associate-*l/ [=>] 80.3	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}\right) \]
      *-inverses [=>] 79.65	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2} \cdot \color{blue}{1}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      *-commutative [=>] 79.65	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{1 \cdot \sqrt{2}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   9. Applied egg-rr 95.67

      \[\leadsto \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right)} - 1\right)}\right) \]

   10. Simplified 80.87

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

       [Start] 95.67	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right)} - 1\right)\right) \]
       expm1-def [=>] 80.97	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right)\right)}\right) \]
       expm1-log1p [=>] 80.87	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\sqrt{\frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}\right) \]
       associate-*r* [=>] 80.87	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)}}\right) \]

   if -6.80000000000000028e176 < A < -5.49999999999999993e-44

   1. Initial program 67.29

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

   2. Simplified 62.86

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

      [Start] 67.29	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 62.57

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

   if -5.49999999999999993e-44 < A < -3.9999999999999999e-85

   1. Initial program 70.6

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

   2. Simplified 67.06

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

      [Start] 70.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 71.82

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

   4. Simplified 71.82

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

      [Start] 71.82	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 71.82	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 71.82	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 74.34

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

   6. Simplified 74.32

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

      [Start] 74.34	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 74.34	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 74.32	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 74.32	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Taylor expanded in B around -inf 86.28

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

   8. Simplified 86.28

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

      [Start] 86.28	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(--1 \cdot \frac{\sqrt{2}}{B}\right) \]
      mul-1-neg [=>] 86.28	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\left(-\frac{\sqrt{2}}{B}\right)}\right) \]

   if -3.9999999999999999e-85 < A < -1.6000000000000001e-140

   1. Initial program 79.27

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

   2. Simplified 74.93

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

      [Start] 79.27	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 73.75

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

   4. Simplified 73.75

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

      [Start] 73.75	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.75	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.75	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 76.51

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

   6. Simplified 76.33

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

      [Start] 76.51	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 76.51	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 76.33	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 76.33	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Applied egg-rr 76.02

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

   8. Simplified 76.02

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

      [Start] 76.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      associate-*l/ [=>] 76.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}\right) \]
      *-inverses [=>] 76.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2} \cdot \color{blue}{1}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      *-commutative [=>] 76.02	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{1 \cdot \sqrt{2}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   9. Taylor expanded in A around -inf 79.23

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

   10. Simplified 79.23

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

       [Start] 79.23	\[ \sqrt{F \cdot \left(2 \cdot A\right)} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
       *-commutative [=>] 79.23	\[ \sqrt{F \cdot \color{blue}{\left(A \cdot 2\right)}} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   if -1.6000000000000001e-140 < A < -5.80000000000000005e-167

   1. Initial program 77.22

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

   2. Simplified 71.45

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

      [Start] 77.22	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 73.46

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

   4. Simplified 73.46

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

      [Start] 73.46	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.46	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.46	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 75.77

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

   6. Simplified 75.74

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

      [Start] 75.77	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 75.77	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 75.74	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 75.74	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Taylor expanded in B around inf 75.27

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

   if -5.80000000000000005e-167 < A < -4.59999999999999991e-303

   1. Initial program 75.91

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

   2. Simplified 70.94

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

      [Start] 75.91	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -4.59999999999999991e-303 < A < 7.80000000000000028e-177

   1. Initial program 76.73

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

   2. Simplified 70.57

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

      [Start] 76.73	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 62.11

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

   4. Simplified 62.11

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

      [Start] 62.11	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 62.11	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 62.11	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 62.23

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

   6. Simplified 61.99

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

      [Start] 62.23	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 62.23	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 61.99	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 61.99	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Applied egg-rr 55.85

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

   8. Simplified 55.85

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

      [Start] 55.85	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      associate-*l/ [=>] 55.85	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}\right) \]
      *-inverses [=>] 55.85	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2} \cdot \color{blue}{1}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      *-commutative [=>] 55.85	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{1 \cdot \sqrt{2}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   9. Applied egg-rr 55.89

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

   if 7.80000000000000028e-177 < A < 5.3000000000000003e-102

   1. Initial program 76.65

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

   2. Simplified 69.72

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

      [Start] 76.65	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 85.58

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

   4. Simplified 85.58

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

      [Start] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r/ [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \color{blue}{\frac{-0.5 \cdot \left(\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)}{C}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \color{blue}{\frac{-0.5}{\frac{C}{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{\color{blue}{{B}^{2} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{\color{blue}{B \cdot B} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(\color{blue}{A \cdot A} - {\left(-1 \cdot A\right)}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 85.58	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B + \left(A \cdot A - {\color{blue}{\left(-A\right)}}^{2}\right)}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 85.83

      \[\leadsto \color{blue}{0 - \frac{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Simplified 85.83

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

      [Start] 85.83	\[ 0 - \frac{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sub0-neg [=>] 85.83	\[ \color{blue}{-\frac{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      mul-1-neg [<=] 85.83	\[ \color{blue}{-1 \cdot \frac{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      metadata-eval [<=] 85.83	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      times-frac [<=] 85.83	\[ \color{blue}{\frac{1 \cdot \sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-1 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      neg-mul-1 [<=] 85.83	\[ \frac{1 \cdot \sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\color{blue}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-lft-identity [=>] 85.83	\[ \frac{\color{blue}{\sqrt{\left(A + \left(A + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, A \cdot A - A \cdot A\right)}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 5.3000000000000003e-102 < A < 1.6999999999999999e92

   1. Initial program 80.72

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

   2. Simplified 76.18

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

      [Start] 80.72	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 73.33

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

   4. Simplified 73.33

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

      [Start] 73.33	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.33	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 73.33	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 73.26

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

   6. Simplified 73.24

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

      [Start] 73.26	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 73.26	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 73.24	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 73.24	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Applied egg-rr 71.92

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

   8. Simplified 71.88

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

      [Start] 71.92	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      associate-*l/ [=>] 71.92	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}\right) \]
      *-inverses [=>] 71.88	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2} \cdot \color{blue}{1}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      *-commutative [=>] 71.88	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{1 \cdot \sqrt{2}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   9. Applied egg-rr 71.78

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

   if 1.6999999999999999e92 < A

   1. Initial program 98.61

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

   2. Simplified 93.31

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

      [Start] 98.61	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 96.68

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

   4. Simplified 96.68

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

      [Start] 96.68	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 96.68	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 96.68	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 96.68

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

   6. Simplified 96.68

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

      [Start] 96.68	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 96.68	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 96.68	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 96.68	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   7. Applied egg-rr 93.58

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

   8. Simplified 92.97

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

      [Start] 93.58	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      associate-*l/ [=>] 93.58	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2} \cdot \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}\right) \]
      *-inverses [=>] 92.97	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2} \cdot \color{blue}{1}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
      *-commutative [=>] 92.97	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{1 \cdot \sqrt{2}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   9. Taylor expanded in B around 0 80.66

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

   10. Simplified 78.16

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

       [Start] 80.66	\[ \sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A - C} + 2 \cdot C\right)} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
       fma-def [=>] 80.66	\[ \sqrt{F \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A - C}, 2 \cdot C\right)}} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
       unpow2 [=>] 80.66	\[ \sqrt{F \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A - C}, 2 \cdot C\right)} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
       associate-/l* [=>] 78.16	\[ \sqrt{F \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{B}{\frac{A - C}{B}}}, 2 \cdot C\right)} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]
       *-commutative [=>] 78.16	\[ \sqrt{F \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, \color{blue}{C \cdot 2}\right)} \cdot \left(-\frac{1 \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

3. Recombined 10 regimes into one program.

4. Final simplification 72.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6.8 \cdot 10^{+176}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ \mathbf{elif}\;A \leq -5.5 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq -4 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot \frac{-\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \mathbf{elif}\;A \leq -5.8 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-303}:\\ \;\;\;\;\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(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 7.8 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2} \cdot \frac{-1}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\right)\\ \mathbf{elif}\;A \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{C \cdot -4} \cdot \sqrt{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	71.89%
Cost	34660

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ t_5 := \sqrt{F \cdot \left(\left(A + C\right) - t_4\right)}\\ t_6 := t_5 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ t_7 := -\sqrt{2}\\ t_8 := \frac{t_7}{t_2}\\ \mathbf{if}\;A \leq -8.8 \cdot 10^{+176}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq -5.5 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(t_4 - C\right) - A\right)\right)} \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-85}:\\ \;\;\;\;t_5 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -1.2 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot t_8\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;t_5 \cdot \frac{t_7}{B}\\ \mathbf{elif}\;A \leq -7.2 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - t_4\right)\right) \cdot \left(F \cdot 2\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 6.8 \cdot 10^{-177}:\\ \;\;\;\;t_5 \cdot \left(\sqrt{2} \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_3}\\ \mathbf{elif}\;A \leq 28:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_8 \cdot \sqrt{F \cdot \mathsf{fma}\left(-0.5, \frac{B}{\frac{A - C}{B}}, C \cdot 2\right)}\\ \end{array} \]

Alternative 2
Error	70.49%
Cost	34532

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := A + \left(C - t_2\right)\\ t_4 := -\sqrt{2}\\ t_5 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_6 := \sqrt{F \cdot \left(\left(A + C\right) - t_2\right)}\\ t_7 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_8 := t_6 \cdot \left(\sqrt{2} \cdot \frac{-1}{t_7}\right)\\ \mathbf{if}\;A \leq -3.8 \cdot 10^{+176}:\\ \;\;\;\;t_6 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ \mathbf{elif}\;A \leq -5.6 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot t_1\right)\right)} \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;A \leq -2.7 \cdot 10^{-84}:\\ \;\;\;\;t_6 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot \frac{t_4}{t_7}\\ \mathbf{elif}\;A \leq -2.9 \cdot 10^{-170}:\\ \;\;\;\;t_6 \cdot \frac{t_4}{B}\\ \mathbf{elif}\;A \leq -2.9 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(t_3 \cdot \left(F \cdot 2\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-177}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;A \leq 2.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_5\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_5}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;t_8\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \frac{-0.5}{\frac{A - C}{B \cdot B}} + F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 3
Error	70.65%
Cost	28576

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := -\sqrt{2}\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ t_5 := \sqrt{F \cdot \left(\left(A + C\right) - t_4\right)}\\ t_6 := t_5 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ t_7 := \frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - t_4\right)\right) \cdot \left(F \cdot 2\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -2.9 \cdot 10^{+177}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq -5.5 \cdot 10^{-44}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;A \leq -2.35 \cdot 10^{-84}:\\ \;\;\;\;t_5 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -4.8 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot \frac{t_2}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-167}:\\ \;\;\;\;t_5 \cdot \frac{t_2}{B}\\ \mathbf{elif}\;A \leq -5.1 \cdot 10^{-235}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq 1.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_3}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \frac{-0.5}{\frac{A - C}{B \cdot B}} + F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 4
Error	70.79%
Cost	28576

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := A + \left(C - t_1\right)\\ t_3 := \sqrt{F \cdot \left(\left(A + C\right) - t_1\right)}\\ t_4 := \mathsf{fma}\left(B, B, t_0\right)\\ t_5 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_6 := t_3 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ t_7 := -\sqrt{2}\\ \mathbf{if}\;A \leq -3.6 \cdot 10^{+176}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq -4.25 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot t_4\right)\right)} \cdot \frac{1}{-t_4}\\ \mathbf{elif}\;A \leq -2.9 \cdot 10^{-84}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot \frac{t_7}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;A \leq -5.6 \cdot 10^{-169}:\\ \;\;\;\;t_3 \cdot \frac{t_7}{B}\\ \mathbf{elif}\;A \leq -2.2 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(t_2 \cdot \left(F \cdot 2\right)\right)}}{t_4}\\ \mathbf{elif}\;A \leq 2.1 \cdot 10^{-162}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_5\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_5}\\ \mathbf{elif}\;A \leq 4.4 \cdot 10^{+90}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \frac{-0.5}{\frac{A - C}{B \cdot B}} + F \cdot \left(C \cdot 2\right)\right)\right)}}{t_4}\\ \end{array} \]

Alternative 5
Error	71.04%
Cost	28248

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := -\sqrt{2}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \sqrt{F \cdot \left(\left(A + C\right) - t_3\right)}\\ t_5 := t_4 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ t_6 := \frac{-\sqrt{t_1 \cdot \left(\left(A + \left(C - t_3\right)\right) \cdot \left(F \cdot 2\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -2.35 \cdot 10^{+177}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;A \leq -4.25 \cdot 10^{-44}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq -2.85 \cdot 10^{-84}:\\ \;\;\;\;t_4 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;A \leq -3.1 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{F \cdot \left(A \cdot 2\right)} \cdot \frac{t_2}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;A \leq -8.4 \cdot 10^{-170}:\\ \;\;\;\;t_4 \cdot \frac{t_2}{B}\\ \mathbf{elif}\;A \leq -4.4 \cdot 10^{-235}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;A \leq 2.05 \cdot 10^{-138}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;A \leq 5.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B}}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+90}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \frac{-0.5}{\frac{A - C}{B \cdot B}} + F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 6
Error	61.93%
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - t_0\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{+65}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_2 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	62.97%
Cost	27208

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-174}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{2}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\right)\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Error	63.47%
Cost	21832

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -0.116:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.16 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A - \left(\sqrt{B \cdot B + {\left(A - C\right)}^{2}} - C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.05 \cdot 10^{-156}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + \left(A + \frac{-0.5}{\frac{C}{B \cdot B}}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-279}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	64.18%
Cost	21528

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := t_1 \cdot \frac{\sqrt{2}}{B}\\ t_3 := \frac{-\sqrt{4 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -360000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-150}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 10
Error	64.25%
Cost	21528

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := t_0 \cdot \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -31000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.05 \cdot 10^{-278}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 11
Error	64.18%
Cost	21268

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ t_3 := t_1 \cdot \frac{\sqrt{2}}{B}\\ t_4 := \frac{-\sqrt{4 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -360000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-152}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-299}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 12
Error	64.21%
Cost	20952

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ t_3 := t_1 \cdot \frac{\sqrt{2}}{B}\\ t_4 := \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{-81}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-150}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-154}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-298}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 13
Error	73.77%
Cost	20556

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 14
Error	76.21%
Cost	15236

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.15 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(F \cdot \left(A \cdot t_0\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 15
Error	81.55%
Cost	14872

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ t_2 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -9.5 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 8.6 \cdot 10^{-252}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 2.6 \cdot 10^{+53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 16
Error	82.12%
Cost	14744

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ t_2 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.82 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -5.4 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-253}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{elif}\;C \leq 1.2 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 17
Error	81.68%
Cost	14744

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ t_2 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -4.4 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-253}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 9.2 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(-16 \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 18
Error	84.3%
Cost	14233

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ t_2 := F \cdot t_0\\ \mathbf{if}\;C \leq -2.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -8.5 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{elif}\;C \leq 3.8 \cdot 10^{-165} \lor \neg \left(C \leq 3 \cdot 10^{-85}\right) \land C \leq 7 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	86.19%
Cost	8844

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -0.26:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1.9 \cdot 10^{-214}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{-160}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	87.85%
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -0.000215:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -2.5 \cdot 10^{-214}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.75 \cdot 10^{-125}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	91.9%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ \mathbf{if}\;A \leq -1.82 \cdot 10^{-84}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	87.88%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + \left(A \cdot C\right) \cdot -4\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -2.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	95.06%
Cost	6980

\[\begin{array}{l} t_0 := \sqrt{A \cdot F}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\frac{2 \cdot t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \]

Alternative 24
Error	95.46%
Cost	6976

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

Alternative 25
Error	96.9%
Cost	6848

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

Reproduce

herbie shell --seed 2023121 
(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))))