ABCF->ab-angle b

Average Error: 52.0 → 38.8

Time: 1.0min

Precision: binary64

Cost: 55372

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t_0\\ t_2 := A - \mathsf{hypot}\left(B, A - C\right)\\ t_3 := F \cdot \left(C + t_2\right)\\ t_4 := \sqrt{t_3}\\ t_5 := B \cdot B + A \cdot A\\ t_6 := A - \sqrt{t_5}\\ \mathbf{if}\;B \leq -6 \cdot 10^{+35}:\\ \;\;\;\;t_4 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot t_0\right)}}{t_0}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0}}{t_0} \cdot \left(\sqrt{\frac{F}{t_6}} \cdot \left(\left(C \cdot \left(1 + A \cdot \sqrt{\frac{1}{t_5}}\right)\right) \cdot -0.5\right) - \sqrt{F \cdot t_6}\right)\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot t_2 + C \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (* F t_0))
        (t_2 (- A (hypot B (- A C))))
        (t_3 (* F (+ C t_2)))
        (t_4 (sqrt t_3))
        (t_5 (+ (* B B) (* A A)))
        (t_6 (- A (sqrt t_5))))
   (if (<= B -6e+35)
     (* t_4 (/ (sqrt 2.0) B))
     (if (<= B -5.5e-90)
       (/ (- (sqrt (* 2.0 (* t_3 t_0)))) t_0)
       (if (<= B -2.3e-115)
         (*
          (/ (sqrt (* 2.0 t_0)) t_0)
          (-
           (* (sqrt (/ F t_6)) (* (* C (+ 1.0 (* A (sqrt (/ 1.0 t_5))))) -0.5))
           (sqrt (* F t_6))))
         (if (<= B 1.3e-141)
           (/ (- (sqrt (* 2.0 (+ (* t_1 t_2) (* C t_1))))) t_0)
           (* t_4 (/ (- (sqrt 2.0)) B))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double t_2 = A - hypot(B, (A - C));
	double t_3 = F * (C + t_2);
	double t_4 = sqrt(t_3);
	double t_5 = (B * B) + (A * A);
	double t_6 = A - sqrt(t_5);
	double tmp;
	if (B <= -6e+35) {
		tmp = t_4 * (sqrt(2.0) / B);
	} else if (B <= -5.5e-90) {
		tmp = -sqrt((2.0 * (t_3 * t_0))) / t_0;
	} else if (B <= -2.3e-115) {
		tmp = (sqrt((2.0 * t_0)) / t_0) * ((sqrt((F / t_6)) * ((C * (1.0 + (A * sqrt((1.0 / t_5))))) * -0.5)) - sqrt((F * t_6)));
	} else if (B <= 1.3e-141) {
		tmp = -sqrt((2.0 * ((t_1 * t_2) + (C * t_1)))) / t_0;
	} else {
		tmp = t_4 * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	t_2 = Float64(A - hypot(B, Float64(A - C)))
	t_3 = Float64(F * Float64(C + t_2))
	t_4 = sqrt(t_3)
	t_5 = Float64(Float64(B * B) + Float64(A * A))
	t_6 = Float64(A - sqrt(t_5))
	tmp = 0.0
	if (B <= -6e+35)
		tmp = Float64(t_4 * Float64(sqrt(2.0) / B));
	elseif (B <= -5.5e-90)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * t_0)))) / t_0);
	elseif (B <= -2.3e-115)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) / t_0) * Float64(Float64(sqrt(Float64(F / t_6)) * Float64(Float64(C * Float64(1.0 + Float64(A * sqrt(Float64(1.0 / t_5))))) * -0.5)) - sqrt(Float64(F * t_6))));
	elseif (B <= 1.3e-141)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_1 * t_2) + Float64(C * t_1))))) / t_0);
	else
		tmp = Float64(t_4 * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(F * N[(C + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(B * B), $MachinePrecision] + N[(A * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(A - N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -6e+35], N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -5.5e-90], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -2.3e-115], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[Sqrt[N[(F / t$95$6), $MachinePrecision]], $MachinePrecision] * N[(N[(C * N[(1.0 + N[(A * N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(F * t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.3e-141], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] + N[(C * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(t$95$4 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -5.99999999999999981e35

   1. Initial program 56.5

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

   2. Simplified 55.3

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

      [Start] 56.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.6

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

   4. Simplified 49.6

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

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

   5. Applied egg-rr 49.5

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

   6. Simplified 49.5

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

      [Start] 49.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [=>] 49.6	\[ \sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 49.6	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [<=] 49.6	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 49.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 49.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 49.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-frac-neg [<=] 49.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Taylor expanded in B around -inf 30.7

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

   if -5.99999999999999981e35 < B < -5.5000000000000003e-90

   1. Initial program 43.9

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

   2. Simplified 37.3

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

      [Start] 43.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -5.5000000000000003e-90 < B < -2.29999999999999985e-115

   1. Initial program 47.4

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

   2. Simplified 42.4

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

      [Start] 47.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 42.9

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

   4. Simplified 42.9

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

      [Start] 42.9	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 42.9	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 42.9	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(A + C\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 42.9	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 42.9	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right)} \cdot -4\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 42.4

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

   6. Simplified 42.2

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

      [Start] 42.4	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [=>] 42.9	\[ \sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 42.9	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [<=] 42.2	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 42.2	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 42.2	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 42.2	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-frac-neg [<=] 42.2	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Taylor expanded in C around 0 53.3

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

   8. Simplified 53.3

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

      [Start] 53.3	\[ \left(\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} + 0.5 \cdot \left(\sqrt{\frac{F}{A - \sqrt{{B}^{2} + {A}^{2}}}} \cdot \left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right)\right)\right)\right) \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 53.3	\[ \left(\sqrt{\left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right) \cdot F} + 0.5 \cdot \left(\sqrt{\frac{F}{A - \sqrt{{B}^{2} + {A}^{2}}}} \cdot \left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right)\right)\right)\right) \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 53.3	\[ \left(\sqrt{\left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right) \cdot F} + 0.5 \cdot \left(\sqrt{\frac{F}{A - \sqrt{{B}^{2} + {A}^{2}}}} \cdot \left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right)\right)\right)\right) \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 53.3	\[ \left(\sqrt{\left(A - \sqrt{B \cdot B + A \cdot A}\right) \cdot F} + \color{blue}{\left(\sqrt{\frac{F}{A - \sqrt{{B}^{2} + {A}^{2}}}} \cdot \left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right)\right)\right) \cdot 0.5}\right) \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 53.3	\[ \left(\sqrt{\left(A - \sqrt{B \cdot B + A \cdot A}\right) \cdot F} + \color{blue}{\sqrt{\frac{F}{A - \sqrt{{B}^{2} + {A}^{2}}}} \cdot \left(\left(C \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{B}^{2} + {A}^{2}}}\right)\right)\right) \cdot 0.5\right)}\right) \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if -2.29999999999999985e-115 < B < 1.30000000000000005e-141

   1. Initial program 52.0

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

   2. Simplified 46.0

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

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

   3. Applied egg-rr 46.1

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

   if 1.30000000000000005e-141 < B

   1. Initial program 51.8

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

   2. Simplified 48.8

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

      [Start] 51.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.5

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

   4. Simplified 46.5

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

      [Start] 46.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 46.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(A + C\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 46.5	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right)} \cdot -4\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 46.2

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

   6. Simplified 46.3

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

      [Start] 46.2	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [=>] 46.5	\[ \sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 46.5	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r- [<=] 46.3	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 46.3	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 46.3	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 46.3	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      distribute-frac-neg [<=] 46.3	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Taylor expanded in B around inf 37.4

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

   8. Simplified 37.4

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

      [Start] 37.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      associate-*r/ [=>] 37.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \]
      mul-1-neg [=>] 37.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\color{blue}{-\sqrt{2}}}{B} \]

3. Recombined 5 regimes into one program.

4. Final simplification 38.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(\sqrt{\frac{F}{A - \sqrt{B \cdot B + A \cdot A}}} \cdot \left(\left(C \cdot \left(1 + A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right) \cdot -0.5\right) - \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right)\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(A - \mathsf{hypot}\left(B, A - C\right)\right) + C \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.5
Cost	34760

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := A - \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{F \cdot \left(C + t_1\right)}\\ t_3 := F \cdot t_0\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+35}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot t_1 + C \cdot t_3\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	38.9
Cost	27984

\[\begin{array}{l} t_0 := F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_1 := \sqrt{t_0}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot t_2\right)}}{t_2}\\ \mathbf{if}\;B \leq -5.4 \cdot 10^{+29}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A \cdot 2\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	38.5
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{+40}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-141}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C - t_0\right)\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	41.0
Cost	27404

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ t_3 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-8}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -7.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(C + A\right) - t_0\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_1}\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{-183}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-169}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(\left(A + A\right) \cdot \left(F \cdot 2\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	40.9
Cost	21528

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := t_0 \cdot \frac{\sqrt{2}}{B}\\ t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -8 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-247}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + A\right) \cdot \left(F \cdot 2\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	41.2
Cost	21084

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \sqrt{\frac{-F}{A}}\\ t_2 := t_0 \cdot \frac{\sqrt{2}}{B}\\ t_3 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-193}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C - \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	47.7
Cost	20492

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C - \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, -A\right)\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot F\right) \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C \cdot 2\right)}\\ \end{array} \]

Alternative 8
Error	48.4
Cost	15632

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{+218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -2.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -5 \cdot 10^{+141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{+194} \lor \neg \left(A \leq 1.1 \cdot 10^{+298}\right):\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 9
Error	51.3
Cost	14737

\[\begin{array}{l} \mathbf{if}\;A \leq -9.2 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 1.15 \cdot 10^{+195} \lor \neg \left(A \leq 1.1 \cdot 10^{+298}\right):\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 10
Error	51.1
Cost	14608

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -6 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -2.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.06 \cdot 10^{+195}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 11
Error	51.1
Cost	14476

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1.16 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{+38}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot -8\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 4.3 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + -0.5 \cdot \left(B \cdot \frac{B}{A - C}\right)\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 12
Error	51.1
Cost	14348

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -3.9 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{+38}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + -0.5 \cdot \left(B \cdot \frac{B}{A - C}\right)\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 13
Error	51.3
Cost	14344

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1.95 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 10^{+195}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + -0.5 \cdot \left(B \cdot \frac{B}{A - C}\right)\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 14
Error	51.5
Cost	13584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{+195}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + -0.5 \cdot \left(B \cdot \frac{B}{A - C}\right)\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 15
Error	54.0
Cost	8980

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{\frac{-F}{A}}\\ t_2 := F \cdot t_0\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-194}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	53.0
Cost	8976

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot 8\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -2.2 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -8 \cdot 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + 0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 7.4 \cdot 10^{-84}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;C \leq 90000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 17
Error	52.4
Cost	8972

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -4 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -3.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C \cdot 2 + -0.5 \cdot \left(B \cdot \frac{B}{A - C}\right)\right)\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	52.7
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -2.15 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.48 \cdot 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 19
Error	53.1
Cost	8588

\[\begin{array}{l} t_0 := -4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1.35 \cdot 10^{+215}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -3.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 8.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(A + \left(C + \left(C - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 20
Error	53.1
Cost	8460

\[\begin{array}{l} t_0 := -4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -4.5 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 4.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 21
Error	53.8
Cost	8337

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -5 \cdot 10^{+217}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3.8 \cdot 10^{-134} \lor \neg \left(A \leq 7.5 \cdot 10^{+176}\right):\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot 8\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	54.0
Cost	8072

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;C \leq -7.2 \cdot 10^{+151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{-210}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot 8\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 4.2 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 23
Error	54.2
Cost	7053

\[\begin{array}{l} \mathbf{if}\;C \leq -2.2 \cdot 10^{+62} \lor \neg \left(C \leq -1.4 \cdot 10^{-271}\right) \land C \leq 1.85 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 24
Error	56.8
Cost	6656

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

Alternative 25
Error	63.3
Cost	6592

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

Reproduce

herbie shell --seed 2023054 
(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))))