ABCF->ab-angle b

Average Error: 52.7 → 38.7

Time: 47.6s

Precision: binary64

Cost: 34648

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \left(C + A\right) - t_0\\ t_4 := \sqrt{F \cdot t_3}\\ t_5 := A + \left(C - t_0\right)\\ \mathbf{if}\;B \leq -9 \cdot 10^{+44}:\\ \;\;\;\;t_4 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_5} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-298}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{elif}\;B \leq 18:\\ \;\;\;\;\frac{-\sqrt{\left(t_5 \cdot \left(F \cdot 2\right)\right) \cdot \left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+48}:\\ \;\;\;\;-\frac{\sqrt{t_1} \cdot \sqrt{t_3 \cdot \left(F \cdot 2\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (hypot B (- A C)))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (fma A (* C -4.0) (* B B)))
        (t_3 (- (+ C A) t_0))
        (t_4 (sqrt (* F t_3)))
        (t_5 (+ A (- C t_0))))
   (if (<= B -9e+44)
     (* t_4 (/ (sqrt 2.0) B))
     (if (<= B -2.65e-30)
       (/ (* (sqrt (* F t_5)) (- (sqrt (* 2.0 t_1)))) t_1)
       (if (<= B 3.35e-298)
         (- (/ (sqrt (* 2.0 (* -4.0 (* (* C A) (* F (+ C C)))))) t_1))
         (if (<= B 3.8e-222)
           (/ (* (sqrt (* 2.0 (* F (* C A)))) (- (sqrt (* A -8.0)))) t_1)
           (if (<= B 18.0)
             (/
              (- (sqrt (* (* t_5 (* F 2.0)) (+ (* -4.0 (* C A)) (* B B)))))
              t_2)
             (if (<= B 2.9e+48)
               (- (/ (* (sqrt t_1) (sqrt (* t_3 (* F 2.0)))) t_2))
               (* 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 = hypot(B, (A - C));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = fma(A, (C * -4.0), (B * B));
	double t_3 = (C + A) - t_0;
	double t_4 = sqrt((F * t_3));
	double t_5 = A + (C - t_0);
	double tmp;
	if (B <= -9e+44) {
		tmp = t_4 * (sqrt(2.0) / B);
	} else if (B <= -2.65e-30) {
		tmp = (sqrt((F * t_5)) * -sqrt((2.0 * t_1))) / t_1;
	} else if (B <= 3.35e-298) {
		tmp = -(sqrt((2.0 * (-4.0 * ((C * A) * (F * (C + C)))))) / t_1);
	} else if (B <= 3.8e-222) {
		tmp = (sqrt((2.0 * (F * (C * A)))) * -sqrt((A * -8.0))) / t_1;
	} else if (B <= 18.0) {
		tmp = -sqrt(((t_5 * (F * 2.0)) * ((-4.0 * (C * A)) + (B * B)))) / t_2;
	} else if (B <= 2.9e+48) {
		tmp = -((sqrt(t_1) * sqrt((t_3 * (F * 2.0)))) / t_2);
	} 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 = hypot(B, Float64(A - C))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_3 = Float64(Float64(C + A) - t_0)
	t_4 = sqrt(Float64(F * t_3))
	t_5 = Float64(A + Float64(C - t_0))
	tmp = 0.0
	if (B <= -9e+44)
		tmp = Float64(t_4 * Float64(sqrt(2.0) / B));
	elseif (B <= -2.65e-30)
		tmp = Float64(Float64(sqrt(Float64(F * t_5)) * Float64(-sqrt(Float64(2.0 * t_1)))) / t_1);
	elseif (B <= 3.35e-298)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(Float64(C * A) * Float64(F * Float64(C + C)))))) / t_1));
	elseif (B <= 3.8e-222)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(C * A)))) * Float64(-sqrt(Float64(A * -8.0)))) / t_1);
	elseif (B <= 18.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_5 * Float64(F * 2.0)) * Float64(Float64(-4.0 * Float64(C * A)) + Float64(B * B))))) / t_2);
	elseif (B <= 2.9e+48)
		tmp = Float64(-Float64(Float64(sqrt(t_1) * sqrt(Float64(t_3 * Float64(F * 2.0)))) / t_2));
	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[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(F * t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(A + N[(C - t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -9e+44], N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.65e-30], N[(N[(N[Sqrt[N[(F * t$95$5), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 3.35e-298], (-N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(N[(C * A), $MachinePrecision] * N[(F * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[B, 3.8e-222], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A * -8.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 18.0], N[((-N[Sqrt[N[(N[(t$95$5 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 2.9e+48], (-N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(t$95$3 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), N[(t$95$4 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -9e44

   1. Initial program 57.9

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

   2. Simplified 56.3

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

      [Start] 57.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 51.1

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

   4. Simplified 51.1

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

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

   5. Applied egg-rr 51.2

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

   6. Simplified 51.1

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

      [Start] 51.2	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 51.2	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)} \]
      associate-*r/ [=>] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      *-rgt-identity [=>] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \]
      distribute-frac-neg [<=] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]
      +-commutative [=>] 51.1	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   7. Taylor expanded in B around -inf 31.3

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

   if -9e44 < B < -2.64999999999999987e-30

   1. Initial program 40.7

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

   2. Simplified 35.8

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

      [Start] 40.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 35.9

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

   4. Simplified 35.9

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

      [Start] 35.9	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 35.9	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 35.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -2.64999999999999987e-30 < B < 3.34999999999999989e-298

   1. Initial program 50.9

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

   2. Simplified 45.2

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

      [Start] 50.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 51.6

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

   4. Simplified 48.8

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

      [Start] 51.6	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 48.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 48.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \color{blue}{\left(F \cdot \left(C - -1 \cdot C\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 48.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 3.34999999999999989e-298 < B < 3.79999999999999997e-222

   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 47.0

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

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

   3. Taylor expanded in C around inf 51.2

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

   4. Simplified 51.2

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

      [Start] 51.2	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 51.2	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.2	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.2	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 51.2	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 49.6

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

   6. Simplified 49.6

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

      [Start] 49.6	\[ \frac{-\sqrt{-8 \cdot A} \cdot \sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.6	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)} \cdot \sqrt{-8 \cdot A}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 51.0	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot C\right) \cdot \left(A + A\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      distribute-rgt-out [<=] 51.0	\[ \frac{-\sqrt{\color{blue}{A \cdot \left(F \cdot C\right) + A \cdot \left(F \cdot C\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      count-2 [=>] 51.0	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(A \cdot \left(F \cdot C\right)\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(F \cdot C\right) \cdot A\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 49.6	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(C \cdot A\right)\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [<=] 49.6	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(A \cdot C\right)}\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.6	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{A \cdot -8}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 3.79999999999999997e-222 < B < 18

   1. Initial program 47.8

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

   2. Simplified 41.9

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

      [Start] 47.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 41.9

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

   if 18 < B < 2.8999999999999999e48

   1. Initial program 42.7

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

   2. Simplified 37.6

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

      [Start] 42.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 36.0

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

   4. Simplified 36.0

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

      [Start] 36.0	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [<=] 36.0	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{{B}^{2}}\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      fma-udef [=>] 36.0	\[ \frac{-\sqrt{\color{blue}{A \cdot \left(C \cdot -4\right) + {B}^{2}}} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*l* [<=] 36.0	\[ \frac{-\sqrt{\color{blue}{\left(A \cdot C\right) \cdot -4} + {B}^{2}} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [<=] 36.0	\[ \frac{-\sqrt{\color{blue}{{B}^{2} + \left(A \cdot C\right) \cdot -4}} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 36.0	\[ \frac{-\sqrt{\color{blue}{B \cdot B} + \left(A \cdot C\right) \cdot -4} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      fma-def [=>] 36.0	\[ \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*l* [=>] 36.0	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 36.0	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 2.8999999999999999e48 < B

   1. Initial program 58.1

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

   2. Simplified 56.7

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

      [Start] 58.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 51.0

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

   4. Simplified 51.0

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

      [Start] 51.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 51.1

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

   6. Simplified 51.1

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

      [Start] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)} \]
      associate-*r/ [=>] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      *-rgt-identity [=>] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right) \]
      distribute-frac-neg [<=] 51.1	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]
      +-commutative [=>] 51.1	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]

   7. Taylor expanded in B around inf 30.6

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

   8. Simplified 30.6

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

      [Start] 30.6	\[ \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      associate-*r/ [=>] 30.6	\[ \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \]
      mul-1-neg [=>] 30.6	\[ \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{\color{blue}{-\sqrt{2}}}{B} \]

3. Recombined 7 regimes into one program.

4. Final simplification 38.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-298}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 18:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+48}:\\ \;\;\;\;-\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(F \cdot 2\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.6
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(\left(C + A\right) - t_0\right)}\\ t_2 := -4 \cdot \left(C \cdot A\right)\\ t_3 := \mathsf{fma}\left(B, B, t_2\right)\\ t_4 := \frac{\sqrt{2 \cdot t_3}}{t_3} \cdot \left(-t_1\right)\\ t_5 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{+45}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-30}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-297}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_5}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(C - t_0\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(t_2 + B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+48}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	38.7
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(\left(C + A\right) - t_0\right)}\\ t_3 := A + \left(C - t_0\right)\\ t_4 := \frac{\sqrt{F \cdot t_3} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{+45}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-298}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\sqrt{\left(t_3 \cdot \left(F \cdot 2\right)\right) \cdot \left(-4 \cdot \left(C \cdot A\right) + B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	40.6
Cost	20876

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -5.6 \cdot 10^{-30}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-297}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C \cdot A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	40.1
Cost	20424

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-127}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	41.9
Cost	20228

\[\begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.46 \cdot 10^{-65}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 6
Error	48.1
Cost	15236

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ \mathbf{if}\;B \leq -4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{B \cdot B + C \cdot C} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-62}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 7
Error	49.5
Cost	14608

\[\begin{array}{l} t_0 := \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 8
Error	49.3
Cost	14608

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(F \cdot C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0 + B \cdot B}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(A \cdot -8\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 3.55 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 9
Error	48.3
Cost	14472

\[\begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 3.75 \cdot 10^{-62}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 10
Error	50.1
Cost	14348

\[\begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{elif}\;B \leq -2.9 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 11
Error	51.0
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -1.1 \cdot 10^{-16}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 12
Error	50.8
Cost	13704

\[\begin{array}{l} t_0 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ t_1 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ \mathbf{if}\;B \leq -3 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-198}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 13
Error	55.2
Cost	8848

\[\begin{array}{l} t_0 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ t_1 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{if}\;A \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-185}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(A \cdot \left(4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 14
Error	55.7
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ t_1 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ \mathbf{if}\;A \leq -1.65 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{-70}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + C\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(A \cdot \left(4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	56.7
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ t_1 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A}{\frac{B \cdot B}{F}}}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-116}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot t_1\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	56.5
Cost	8584

\[\begin{array}{l} t_0 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ t_1 := -4 \cdot \left(C \cdot A\right) + B \cdot B\\ \mathbf{if}\;A \leq -7.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;A \leq 10^{-238}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \end{array} \]

Alternative 17
Error	58.7
Cost	8452

\[\begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A}{\frac{B \cdot B}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \end{array} \]

Alternative 18
Error	60.4
Cost	8072

\[\begin{array}{l} \mathbf{if}\;B \leq -3.5 \cdot 10^{-9}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A}{\frac{B \cdot B}{F}}}\\ \mathbf{elif}\;B \leq 5.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \end{array} \]

Alternative 19
Error	60.2
Cost	8072

\[\begin{array}{l} \mathbf{if}\;B \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A}{\frac{B \cdot B}{F}}}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(A \cdot \left(4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{A}{B \cdot B}}\\ \end{array} \]

Alternative 20
Error	61.2
Cost	6976

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

Alternative 21
Error	62.0
Cost	6848

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

Reproduce

herbie shell --seed 2023067 
(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))))