ABCF->ab-angle b

Average Error: 52.5 → 41.2

Time: 43.2s

Precision: binary64

Cost: 21268

Math

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

↓

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := -\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ t_3 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -4.4 \cdot 10^{-109}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{2 \cdot \left(\left(A \cdot \sqrt{2}\right) \cdot \sqrt{C \cdot \left(F \cdot -2\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + A\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (* -4.0 (* C A)))
        (t_1 (+ (* B B) t_0))
        (t_2
         (-
          (/
           (sqrt (* -8.0 (* A (* (* F C) (+ C C)))))
           (+ (* B B) (* A (* C -4.0))))))
        (t_3 (sqrt (* F (+ C (- A (hypot B (- A C))))))))
   (if (<= B -4.4e-109)
     (* t_3 (/ (sqrt 2.0) B))
     (if (<= B 1e-301)
       t_2
       (if (<= B 1.05e-160)
         (/ (* 2.0 (* (* A (sqrt 2.0)) (sqrt (* C (* F -2.0))))) t_1)
         (if (<= B 7.6e-116)
           t_2
           (if (<= B 2.8e-67)
             (/
              (*
               (sqrt (* F (+ A A)))
               (- (sqrt (* 2.0 (fma A (* C -4.0) (* B B))))))
              t_0)
             (if (<= B 3.7e-47)
               (/ (- (sqrt (* 2.0 (* (* F t_1) (+ A (+ C A)))))) t_1)
               (* t_3 (/ (- (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 = -4.0 * (C * A);
	double t_1 = (B * B) + t_0;
	double t_2 = -(sqrt((-8.0 * (A * ((F * C) * (C + C))))) / ((B * B) + (A * (C * -4.0))));
	double t_3 = sqrt((F * (C + (A - hypot(B, (A - C))))));
	double tmp;
	if (B <= -4.4e-109) {
		tmp = t_3 * (sqrt(2.0) / B);
	} else if (B <= 1e-301) {
		tmp = t_2;
	} else if (B <= 1.05e-160) {
		tmp = (2.0 * ((A * sqrt(2.0)) * sqrt((C * (F * -2.0))))) / t_1;
	} else if (B <= 7.6e-116) {
		tmp = t_2;
	} else if (B <= 2.8e-67) {
		tmp = (sqrt((F * (A + A))) * -sqrt((2.0 * fma(A, (C * -4.0), (B * B))))) / t_0;
	} else if (B <= 3.7e-47) {
		tmp = -sqrt((2.0 * ((F * t_1) * (A + (C + A))))) / t_1;
	} else {
		tmp = t_3 * (-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 = Float64(-4.0 * Float64(C * A))
	t_1 = Float64(Float64(B * B) + t_0)
	t_2 = Float64(-Float64(sqrt(Float64(-8.0 * Float64(A * Float64(Float64(F * C) * Float64(C + C))))) / Float64(Float64(B * B) + Float64(A * Float64(C * -4.0)))))
	t_3 = sqrt(Float64(F * Float64(C + Float64(A - hypot(B, Float64(A - C))))))
	tmp = 0.0
	if (B <= -4.4e-109)
		tmp = Float64(t_3 * Float64(sqrt(2.0) / B));
	elseif (B <= 1e-301)
		tmp = t_2;
	elseif (B <= 1.05e-160)
		tmp = Float64(Float64(2.0 * Float64(Float64(A * sqrt(2.0)) * sqrt(Float64(C * Float64(F * -2.0))))) / t_1);
	elseif (B <= 7.6e-116)
		tmp = t_2;
	elseif (B <= 2.8e-67)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + A))) * Float64(-sqrt(Float64(2.0 * fma(A, Float64(C * -4.0), Float64(B * B)))))) / t_0);
	elseif (B <= 3.7e-47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(A + Float64(C + A)))))) / t_1);
	else
		tmp = Float64(t_3 * 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[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[Sqrt[N[(-8.0 * N[(A * N[(N[(F * C), $MachinePrecision] * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(B * B), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$3 = N[Sqrt[N[(F * N[(C + N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -4.4e-109], N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1e-301], t$95$2, If[LessEqual[B, 1.05e-160], N[(N[(2.0 * N[(N[(A * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 7.6e-116], t$95$2, If[LessEqual[B, 2.8e-67], N[(N[(N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 3.7e-47], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(A + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(t$95$3 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -4.3999999999999999e-109

   1. Initial program 52.4

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

   2. Simplified 49.2

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

      [Start] 52.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.0

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

   4. Simplified 46.0

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

      [Start] 46.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 46.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 46.0

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

   6. Simplified 46.0

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

      [Start] 46.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-lft-neg-out [=>] 46.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)} \]
      associate-*r/ [=>] 46.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot 1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]
      *-rgt-identity [=>] 46.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-frac-neg [<=] 46.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around -inf 36.6

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

   if -4.3999999999999999e-109 < B < 1.00000000000000007e-301 or 1.05e-160 < B < 7.6000000000000003e-116

   1. Initial program 51.6

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

   2. Simplified 46.0

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

      [Start] 51.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.0

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

   4. Taylor expanded in A around inf 51.2

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

   5. Simplified 51.2

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

      [Start] 51.2	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]
      *-commutative [=>] 51.2	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)}\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]
      *-commutative [=>] 51.2	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(C - -1 \cdot C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]
      mul-1-neg [=>] 51.2	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

   if 1.00000000000000007e-301 < B < 1.05e-160

   1. Initial program 54.7

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

   2. Simplified 53.9

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

      [Start] 54.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 56.0

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

   4. Simplified 56.0

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

      [Start] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{{B}^{2} + {C}^{2}} + -1 \cdot \left(\left(A \cdot C\right) \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{\color{blue}{B \cdot B} + {C}^{2}} + -1 \cdot \left(\left(A \cdot C\right) \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{B \cdot B + \color{blue}{C \cdot C}} + -1 \cdot \left(\left(A \cdot C\right) \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{B \cdot B + C \cdot C} + \color{blue}{\left(-\left(A \cdot C\right) \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}\right)}\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{B \cdot B + C \cdot C} + \left(-\left(A \cdot C\right) \cdot \sqrt{\frac{1}{\color{blue}{B \cdot B} + {C}^{2}}}\right)\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 56.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \left(\sqrt{B \cdot B + C \cdot C} + \left(-\left(A \cdot C\right) \cdot \sqrt{\frac{1}{B \cdot B + \color{blue}{C \cdot C}}}\right)\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around -inf 57.2

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

   6. Simplified 57.2

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

      [Start] 57.2	\[ \frac{--2 \cdot \left(\left(\sqrt{2} \cdot A\right) \cdot \sqrt{C \cdot \left(F \cdot \left(-1 \cdot \left(C \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}\right) - 1\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-neg [=>] 57.2	\[ \frac{--2 \cdot \left(\left(\sqrt{2} \cdot A\right) \cdot \sqrt{C \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(-1, C \cdot \sqrt{\frac{1}{{B}^{2} + {C}^{2}}}, -1\right)}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 57.2	\[ \frac{--2 \cdot \left(\left(\sqrt{2} \cdot A\right) \cdot \sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-1, C \cdot \sqrt{\frac{1}{\color{blue}{B \cdot B} + {C}^{2}}}, -1\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 57.2	\[ \frac{--2 \cdot \left(\left(\sqrt{2} \cdot A\right) \cdot \sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-1, C \cdot \sqrt{\frac{1}{B \cdot B + \color{blue}{C \cdot C}}}, -1\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 57.2	\[ \frac{--2 \cdot \left(\left(\sqrt{2} \cdot A\right) \cdot \sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-1, C \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}, \color{blue}{-1}\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Taylor expanded in C around inf 51.1

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

   if 7.6000000000000003e-116 < B < 2.8000000000000001e-67

   1. Initial program 46.9

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

   2. Simplified 40.3

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

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

   3. Applied egg-rr 41.7

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

   4. Simplified 41.7

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

      [Start] 41.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 41.7	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 41.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Taylor expanded in C around inf 54.4

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

   6. Simplified 54.4

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

      [Start] 54.4	\[ \frac{-\sqrt{\left(A - -1 \cdot A\right) \cdot F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 54.4	\[ \frac{-\sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 54.4	\[ \frac{-\sqrt{F \cdot \left(A - \color{blue}{\left(-A\right)}\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   7. Taylor expanded in B around 0 56.4

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

   8. Simplified 56.4

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

      [Start] 56.4	\[ \frac{-\sqrt{F \cdot \left(A - \left(-A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{-4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 56.4	\[ \frac{-\sqrt{F \cdot \left(A - \left(-A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\color{blue}{\left(A \cdot C\right) \cdot -4}} \]

   if 2.8000000000000001e-67 < B < 3.7e-47

   1. Initial program 47.0

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

   2. Simplified 46.7

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

      [Start] 47.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 55.9

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

   4. Simplified 55.9

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

      [Start] 55.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - -1 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 55.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \color{blue}{\left(-A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 3.7e-47 < B

   1. Initial program 53.5

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

   2. Simplified 51.3

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

      [Start] 53.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.9

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

   4. Simplified 46.9

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

      [Start] 46.9	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.9	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 46.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 46.9

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

   6. Simplified 46.9

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

      [Start] 46.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-lft-neg-out [=>] 46.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)} \]
      associate-*r/ [=>] 46.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot 1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]
      *-rgt-identity [=>] 46.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-frac-neg [<=] 46.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around 0 33.0

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

   8. Simplified 33.0

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

      [Start] 33.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      mul-1-neg [=>] 33.0	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 41.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.4 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 10^{-301}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{2 \cdot \left(\left(A \cdot \sqrt{2}\right) \cdot \sqrt{C \cdot \left(F \cdot -2\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-116}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + A\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.4
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{-15}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot \left(\left(t_0 - A\right) - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	38.1
Cost	27720

\[\begin{array}{l} t_0 := F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_1 := \sqrt{t_0}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-15}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot t_2\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	38.1
Cost	21448

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{-15}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, t_2\right) \cdot \left(\left(A + \left(C - t_0\right)\right) \cdot \left(F \cdot 2\right)\right)}}{B \cdot B + t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	39.8
Cost	21192

\[\begin{array}{l} t_0 := F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_1 := \sqrt{t_0}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-107}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	40.8
Cost	20688

\[\begin{array}{l} t_0 := -\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-106}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-290}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-159}:\\ \;\;\;\;\frac{2 \cdot \left(\left(A \cdot \sqrt{2}\right) \cdot \sqrt{C \cdot \left(F \cdot -2\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	42.5
Cost	20228

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := -\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{if}\;B \leq -6 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{2 \cdot \left(\left(A \cdot \sqrt{2}\right) \cdot \sqrt{C \cdot \left(F \cdot -2\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 7
Error	47.7
Cost	14540

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := -\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{if}\;B \leq -1.26 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot 2\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{2 \cdot \left(\left(A \cdot \sqrt{2}\right) \cdot \sqrt{C \cdot \left(F \cdot -2\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{-21}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 8
Error	47.9
Cost	14276

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{-103}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot 2\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-108}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 9
Error	48.7
Cost	13836

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -5.6 \cdot 10^{-102}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-105}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-23}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 10
Error	52.2
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.45 \cdot 10^{-24}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-216}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - C \cdot -2\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 11
Error	52.5
Cost	8324

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq 5 \cdot 10^{-92}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + A\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{B \cdot B + A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 12
Error	56.2
Cost	8205

\[\begin{array}{l} t_0 := B \cdot B + A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;C \leq -1.9 \cdot 10^{-237} \lor \neg \left(C \leq 5.4 \cdot 10^{-81}\right) \land C \leq 1.5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 13
Error	55.9
Cost	7944

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;A \leq -2 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{F \cdot A} \cdot \frac{2}{B}\\ \mathbf{elif}\;A \leq -2.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{B \cdot B + t_0}\\ \end{array} \]

Alternative 14
Error	54.2
Cost	7940

\[\begin{array}{l} t_0 := B \cdot B + A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;A \leq 1.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	58.1
Cost	7556

\[\begin{array}{l} \mathbf{if}\;A \leq -2.3 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{F \cdot A} \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 16
Error	62.1
Cost	6848

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

Alternative 17
Error	62.2
Cost	6848

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

Reproduce

herbie shell --seed 2023060 
(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))))