ABCF->ab-angle a

Average Error: 52.1 → 36.6

Time: 58.0s

Precision: binary64

Cost: 27736

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + F \cdot \left(2 \cdot C\right)\right)\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;B \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;t_1 \cdot \frac{t_0}{-B}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_4} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot C\right)}\right)}{t_4}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-94}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{t_4}} \cdot \left(-\sqrt{2 \cdot \left(A + A\right)}\right)\\ \mathbf{elif}\;B \leq 0.036:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt F)))
        (t_1 (sqrt (* 2.0 (+ A (+ C (hypot B (- A C)))))))
        (t_2 (fma B B (* C (* A -4.0))))
        (t_3
         (/
          (-
           (sqrt
            (* 2.0 (* t_2 (+ (* F (* (* B B) (/ -0.5 A))) (* F (* 2.0 C)))))))
          t_2))
        (t_4 (fma -4.0 (* A C) (* B B))))
   (if (<= B -2.8e-60)
     (* t_1 (/ t_0 (- B)))
     (if (<= B -1.65e-260)
       (/ (* (sqrt (* F t_4)) (- (sqrt (* 2.0 (* 2.0 C))))) t_4)
       (if (<= B 3.2e-239)
         (/
          (- (pow (* (* A (* F (* A C))) -16.0) 0.5))
          (+ (* B B) (* -4.0 (* A C))))
         (if (<= B 8e-94)
           t_3
           (if (<= B 1.45e-48)
             (* (/ (sqrt F) (sqrt t_4)) (- (sqrt (* 2.0 (+ A A)))))
             (if (<= B 0.036)
               t_3
               (* t_1 (/ t_0 (fma -2.0 (* C (/ A B)) 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 = -sqrt(F);
	double t_1 = sqrt((2.0 * (A + (C + hypot(B, (A - C))))));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double t_3 = -sqrt((2.0 * (t_2 * ((F * ((B * B) * (-0.5 / A))) + (F * (2.0 * C)))))) / t_2;
	double t_4 = fma(-4.0, (A * C), (B * B));
	double tmp;
	if (B <= -2.8e-60) {
		tmp = t_1 * (t_0 / -B);
	} else if (B <= -1.65e-260) {
		tmp = (sqrt((F * t_4)) * -sqrt((2.0 * (2.0 * C)))) / t_4;
	} else if (B <= 3.2e-239) {
		tmp = -pow(((A * (F * (A * C))) * -16.0), 0.5) / ((B * B) + (-4.0 * (A * C)));
	} else if (B <= 8e-94) {
		tmp = t_3;
	} else if (B <= 1.45e-48) {
		tmp = (sqrt(F) / sqrt(t_4)) * -sqrt((2.0 * (A + A)));
	} else if (B <= 0.036) {
		tmp = t_3;
	} else {
		tmp = t_1 * (t_0 / fma(-2.0, (C * (A / B)), B));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(-sqrt(F))
	t_1 = sqrt(Float64(2.0 * Float64(A + Float64(C + hypot(B, Float64(A - C))))))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_3 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(Float64(F * Float64(Float64(B * B) * Float64(-0.5 / A))) + Float64(F * Float64(2.0 * C))))))) / t_2)
	t_4 = fma(-4.0, Float64(A * C), Float64(B * B))
	tmp = 0.0
	if (B <= -2.8e-60)
		tmp = Float64(t_1 * Float64(t_0 / Float64(-B)));
	elseif (B <= -1.65e-260)
		tmp = Float64(Float64(sqrt(Float64(F * t_4)) * Float64(-sqrt(Float64(2.0 * Float64(2.0 * C))))) / t_4);
	elseif (B <= 3.2e-239)
		tmp = Float64(Float64(-(Float64(Float64(A * Float64(F * Float64(A * C))) * -16.0) ^ 0.5)) / Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))));
	elseif (B <= 8e-94)
		tmp = t_3;
	elseif (B <= 1.45e-48)
		tmp = Float64(Float64(sqrt(F) / sqrt(t_4)) * Float64(-sqrt(Float64(2.0 * Float64(A + A)))));
	elseif (B <= 0.036)
		tmp = t_3;
	else
		tmp = Float64(t_1 * Float64(t_0 / fma(-2.0, Float64(C * Float64(A / B)), 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[Sqrt[F], $MachinePrecision])}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(N[(F * N[(N[(B * B), $MachinePrecision] * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.8e-60], N[(t$95$1 * N[(t$95$0 / (-B)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.65e-260], N[(N[(N[Sqrt[N[(F * t$95$4), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 3.2e-239], N[((-N[Power[N[(N[(A * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision], 0.5], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 8e-94], t$95$3, If[LessEqual[B, 1.45e-48], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 0.036], t$95$3, N[(t$95$1 * N[(t$95$0 / N[(-2.0 * N[(C * N[(A / B), $MachinePrecision]), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -2.8000000000000002e-60

   1. Initial program 52.6

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

   2. Simplified 50.1

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

      [Start] 52.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 45.9

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

   4. Simplified 45.8

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

      [Start] 45.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 45.9	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 45.9	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 45.9	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 45.9	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 45.8	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 45.9

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

   6. Simplified 45.8

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

      [Start] 45.9	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \]
      associate-*r/ [=>] 45.9	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
      *-rgt-identity [=>] 45.9	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\color{blue}{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
      distribute-frac-neg [=>] 45.9	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)} \]
      distribute-rgt-neg-out [=>] 45.9	\[ \color{blue}{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
      associate-*r/ [=>] 45.9	\[ -\color{blue}{\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   7. Applied egg-rr 42.6

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

   8. Simplified 41.4

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

      [Start] 42.6	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\right) \]
      associate-*l/ [=>] 42.6	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{\sqrt{F} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}} \]
      *-inverses [=>] 41.4	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot \color{blue}{1}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   9. Taylor expanded in B around -inf 27.2

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

   10. Simplified 27.2

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

       [Start] 27.2	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{-1 \cdot B} \]
       mul-1-neg [=>] 27.2	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{\color{blue}{-B}} \]

   if -2.8000000000000002e-60 < B < -1.6499999999999999e-260

   1. Initial program 51.3

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

   2. Simplified 46.1

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

      [Start] 51.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 42.1

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

   4. Simplified 41.5

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

      [Start] 42.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 42.1	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 42.1	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 42.1	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 42.1	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 41.5	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Taylor expanded in A around -inf 48.9

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

   if -1.6499999999999999e-260 < B < 3.1999999999999999e-239

   1. Initial program 54.0

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

   2. Simplified 54.0

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

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

   3. Taylor expanded in B around 0 55.0

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

   4. Simplified 54.9

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

      [Start] 55.0	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 54.9	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 54.9	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 50.1

      \[\leadsto \frac{-\color{blue}{{\left(\left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right) \cdot -16\right)}^{0.5}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 3.1999999999999999e-239 < B < 7.9999999999999996e-94 or 1.4500000000000001e-48 < B < 0.0359999999999999973

   1. Initial program 48.6

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

   2. Simplified 44.1

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

      [Start] 48.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 49.4

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

   4. Simplified 49.4

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

      [Start] 49.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 49.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [=>] 49.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 49.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 49.4

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

   if 7.9999999999999996e-94 < B < 1.4500000000000001e-48

   1. Initial program 47.2

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

   2. Simplified 41.7

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

      [Start] 47.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 38.2

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

   4. Simplified 37.7

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

      [Start] 38.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 38.2	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 38.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 37.7	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 38.2

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

   6. Simplified 37.7

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

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

   7. Applied egg-rr 39.1

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

   8. Simplified 38.9

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

      [Start] 39.1	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\right) \]
      associate-*l/ [=>] 39.1	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{\sqrt{F} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}} \]
      *-inverses [=>] 38.9	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot \color{blue}{1}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   9. Taylor expanded in C around -inf 54.2

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

   if 0.0359999999999999973 < B

   1. Initial program 54.3

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

   2. Simplified 52.5

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

      [Start] 54.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 48.2

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

   4. Simplified 48.2

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

      [Start] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 48.2	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 48.2	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 48.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 48.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 48.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 48.2

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

   6. Simplified 48.2

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

      [Start] 48.2	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right) \]
      associate-*r/ [=>] 48.2	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
      *-rgt-identity [=>] 48.2	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\color{blue}{-\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
      distribute-frac-neg [=>] 48.2	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)} \]
      distribute-rgt-neg-out [=>] 48.2	\[ \color{blue}{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]
      associate-*r/ [=>] 48.2	\[ -\color{blue}{\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   7. Applied egg-rr 43.5

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

   8. Simplified 42.2

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

      [Start] 43.5	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}\right) \]
      associate-*l/ [=>] 43.5	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{\sqrt{F} \cdot \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}} \]
      *-inverses [=>] 42.2	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot \color{blue}{1}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   9. Taylor expanded in A around 0 25.1

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

   10. Simplified 23.9

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

       [Start] 25.1	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{-2 \cdot \frac{A \cdot C}{B} + B} \]
       fma-def [=>] 25.1	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{\color{blue}{\mathsf{fma}\left(-2, \frac{A \cdot C}{B}, B\right)}} \]
       associate-/l* [=>] 23.9	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{\mathsf{fma}\left(-2, \color{blue}{\frac{A}{\frac{B}{C}}}, B\right)} \]
       associate-/r/ [=>] 23.9	\[ -\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{F} \cdot 1}{\mathsf{fma}\left(-2, \color{blue}{\frac{A}{B} \cdot C}, B\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 36.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{F}}{-B}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot C\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + F \cdot \left(2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \left(-\sqrt{2 \cdot \left(A + A\right)}\right)\\ \mathbf{elif}\;B \leq 0.036:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + F \cdot \left(2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	32.7
Cost	155788

\[\begin{array}{l} t_0 := C \cdot \left(A \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := -\sqrt{F}\\ t_3 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_4 := {B}^{2} + t_0\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_4}\\ t_6 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;t_5 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;t_6 \cdot \frac{t_2}{\sqrt{t_3}}\\ \mathbf{elif}\;t_5 \leq 5 \cdot 10^{-59}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{t_6 \cdot \left(-\sqrt{F \cdot t_3}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \frac{t_2}{2 \cdot \left(C \cdot \frac{A}{B}\right) - B}\\ \end{array} \]

Alternative 2
Error	31.9
Cost	27848

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \sqrt{2 \cdot t_1}\\ \mathbf{if}\;B \leq -62:\\ \;\;\;\;t_2 \cdot \frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{t_1}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 3
Error	36.7
Cost	27736

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := -\sqrt{F}\\ t_3 := F \cdot \left(2 \cdot C\right)\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + t_3\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-36}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{-B}\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot t_3\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-239}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \left(-\sqrt{2 \cdot \left(A + A\right)}\right)\\ \mathbf{elif}\;B \leq 0.24:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 4
Error	34.9
Cost	27728

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := -\sqrt{F}\\ t_2 := \sqrt{2 \cdot \left(A + \left(C + t_0\right)\right)}\\ t_3 := \frac{\sqrt{-8 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{C + \left(A + t_0\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{-54}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{-B}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-232}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 5
Error	34.3
Cost	27592

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{-43}:\\ \;\;\;\;t_1 \cdot \frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-14}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{F \cdot \left(A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 6
Error	36.7
Cost	27284

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := -\sqrt{F}\\ t_3 := F \cdot \left(2 \cdot C\right)\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + t_3\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-38}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{-B}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot t_3\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \left(-\sqrt{2 \cdot \left(A + A\right)}\right)\\ \mathbf{elif}\;B \leq 0.31:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{B}\\ \end{array} \]

Alternative 7
Error	36.6
Cost	21904

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := -\sqrt{F}\\ t_3 := F \cdot \left(2 \cdot C\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-37}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{-B}\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-262}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot t_3\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 0.18:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \frac{-0.5}{A}\right) + t_3\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{B}\\ \end{array} \]

Alternative 8
Error	36.7
Cost	21528

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\sqrt{F}\\ t_2 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{t_3}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{-B}\\ \mathbf{elif}\;B \leq -2.26 \cdot 10^{-262}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-234}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ \mathbf{elif}\;B \leq 0.039:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{B}\\ \end{array} \]

Alternative 9
Error	36.6
Cost	21328

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := -\sqrt{F}\\ t_2 := \sqrt{2 \cdot \left(A + \left(C + t_0\right)\right)}\\ t_3 := F \cdot \left(A \cdot C\right)\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{-B}\\ \mathbf{elif}\;B \leq -1.85 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 1.62 \cdot 10^{-239}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_3\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 0.035:\\ \;\;\;\;\frac{-\sqrt{\left(-8 \cdot t_3\right) \cdot \left(C + \left(A + t_0\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_1}{B}\\ \end{array} \]

Alternative 10
Error	53.2
Cost	21280

\[\begin{array}{l} t_0 := A + \sqrt{B \cdot B + A \cdot A}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_1}\\ t_3 := F \cdot \left(B \cdot B\right)\\ t_4 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot t_3\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-202}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 0.034:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot t_0}\\ \end{array} \]

Alternative 11
Error	46.4
Cost	21280

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ t_2 := F \cdot \left(B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot t_2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 12
Error	46.4
Cost	21216

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ t_2 := F \cdot \left(B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot t_2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-238}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{F}}{B}\\ \end{array} \]

Alternative 13
Error	38.2
Cost	21216

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ t_2 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_3 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{-35}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.56 \cdot 10^{-202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-239}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{F}}{B}\\ \end{array} \]

Alternative 14
Error	38.2
Cost	21216

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_1}\\ t_3 := \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_4 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -7 \cdot 10^{-37}:\\ \;\;\;\;t_3 \cdot \frac{t_0}{-B}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-199}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-305}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-94}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{t_0}{B}\\ \end{array} \]

Alternative 15
Error	54.2
Cost	16352

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ t_2 := F \cdot \left(B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_0}\\ t_4 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot t_2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.56 \cdot 10^{-202}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.9 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 0.034:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	54.0
Cost	15776

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ t_2 := F \cdot \left(B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_0}\\ t_4 := \frac{-\sqrt{2 \cdot \mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(C \cdot F\right)\right), 4 \cdot \left(C \cdot t_2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -7.6 \cdot 10^{-37}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-198}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-238}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.65 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 0.035:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 17
Error	53.2
Cost	15120

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ \mathbf{if}\;A \leq -1.05 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.62 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{+21}:\\ \;\;\;\;\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;A \leq 2.7 \cdot 10^{+68}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 18
Error	54.5
Cost	15004

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ t_2 := F \cdot t_0\\ t_3 := \frac{\sqrt{A \cdot \left(\left(C \cdot F\right) \cdot -16\right)} \cdot \left(-\sqrt{A}\right)}{t_0}\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -3.7 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-292}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-235}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	53.5
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -4.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.55 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \end{array} \]

Alternative 20
Error	55.2
Cost	7945

\[\begin{array}{l} \mathbf{if}\;A \leq -2.5 \cdot 10^{-77} \lor \neg \left(A \leq 8.1 \cdot 10^{+65}\right):\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 21
Error	55.4
Cost	7945

\[\begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-304} \lor \neg \left(F \leq 9 \cdot 10^{+59}\right):\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 22
Error	54.8
Cost	7940

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq 4.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \end{array} \]

Alternative 23
Error	55.4
Cost	7876

\[\begin{array}{l} \mathbf{if}\;A \leq -2.8 \cdot 10^{-77}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 24
Error	57.0
Cost	6656

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

Reproduce

herbie shell --seed 2023057 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))