ABCF->ab-angle a

Average Error: 52.4 → 30.4

Time: 57.1s

Precision: binary64

Cost: 27848

Math

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

↓

\[\begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{t_0} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ C (+ A (hypot B (- A C))))))
   (if (<= B -5.2e+38)
     (* (sqrt t_0) (* (sqrt F) (/ (sqrt 2.0) B)))
     (if (<= B 3.5e+49)
       (/
        (* (sqrt (* t_0 2.0)) (- (sqrt (* F (fma -4.0 (* C A) (* B B))))))
        (+ (* B B) (* -4.0 (* C A))))
       (* (/ (- (sqrt 2.0)) B) (* (sqrt F) (sqrt (+ C (hypot C 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 = C + (A + hypot(B, (A - C)));
	double tmp;
	if (B <= -5.2e+38) {
		tmp = sqrt(t_0) * (sqrt(F) * (sqrt(2.0) / B));
	} else if (B <= 3.5e+49) {
		tmp = (sqrt((t_0 * 2.0)) * -sqrt((F * fma(-4.0, (C * A), (B * B))))) / ((B * B) + (-4.0 * (C * A)));
	} else {
		tmp = (-sqrt(2.0) / B) * (sqrt(F) * sqrt((C + hypot(C, 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(C + Float64(A + hypot(B, Float64(A - C))))
	tmp = 0.0
	if (B <= -5.2e+38)
		tmp = Float64(sqrt(t_0) * Float64(sqrt(F) * Float64(sqrt(2.0) / B)));
	elseif (B <= 3.5e+49)
		tmp = Float64(Float64(sqrt(Float64(t_0 * 2.0)) * Float64(-sqrt(Float64(F * fma(-4.0, Float64(C * A), Float64(B * B)))))) / Float64(Float64(B * B) + Float64(-4.0 * Float64(C * A))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * Float64(sqrt(F) * sqrt(Float64(C + hypot(C, 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[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -5.2e+38], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.5e+49], N[(N[(N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -5.1999999999999998e38

   1. Initial program 56.4

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

   2. Simplified 55.1

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

      [Start] 56.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.5

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

   4. Simplified 49.4

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

      [Start] 49.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 49.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+l+ [=>] 49.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 47.3

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

   6. Applied egg-rr 62.9

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

   7. Simplified 46.0

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

      [Start] 62.9	\[ e^{\mathsf{log1p}\left(\frac{-\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)} - 1 \]
      expm1-def [=>] 52.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 47.3	\[ \color{blue}{\frac{-\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      distribute-lft-neg-in [=>] 47.3	\[ \frac{\color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [<=] 46.0	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      distribute-rgt-neg-in [=>] 46.0	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{F}\right)\right)} \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*l* [=>] 46.0	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{F}\right) \cdot \frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)} \]

   8. Taylor expanded in B around -inf 22.0

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

   9. Simplified 22.0

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

      [Start] 22.0	\[ \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{F}\right) \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\right) \]
      mul-1-neg [=>] 22.0	\[ \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{F}\right) \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)}\right) \]
      distribute-neg-frac [=>] 22.0	\[ \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{F}\right) \cdot \color{blue}{\frac{-\sqrt{2}}{B}}\right) \]

   if -5.1999999999999998e38 < B < 3.49999999999999975e49

   1. Initial program 48.8

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

   2. Simplified 42.9

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

      [Start] 48.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 38.2

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

   4. Simplified 38.1

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

      [Start] 38.2	\[ \frac{-\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 38.2	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.2	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [=>] 38.8	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 38.8	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+l+ [=>] 38.1	\[ \frac{-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 38.1

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

   if 3.49999999999999975e49 < B

   1. Initial program 57.6

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

   2. Simplified 55.9

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

      [Start] 57.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 54.1

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

   4. Simplified 54.1

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

      [Start] 54.1	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 54.1	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 54.1	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 54.1	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      *-commutative [=>] 54.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]
      +-commutative [=>] 54.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)} \]
      unpow2 [=>] 54.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)} \]
      unpow2 [=>] 54.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)} \]

   5. Applied egg-rr 51.8

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

   6. Simplified 19.1

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

      [Start] 51.8	\[ \frac{-\sqrt{2}}{B} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right) \]
      hypot-def [=>] 19.1	\[ \frac{-\sqrt{2}}{B} \cdot \left(\sqrt{C + \color{blue}{\mathsf{hypot}\left(C, B\right)}} \cdot \sqrt{F}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 30.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot 2} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	32.9
Cost	27724

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C + \left(A + t_0\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{t_1} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq -5.7 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C + t_0\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot 2} \cdot \left(-\sqrt{F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 2
Error	33.6
Cost	27592

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -9 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{t_1} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot 2} \cdot \left(-\sqrt{F \cdot t_0}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 3
Error	39.5
Cost	26700

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{B \cdot B + C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.82 \cdot 10^{-19}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 4
Error	33.8
Cost	26628

\[\begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-19}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(C, B\right)}\right)\\ \end{array} \]

Alternative 5
Error	44.9
Cost	21132

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := -\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq -9.6 \cdot 10^{-279}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{-1}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \]

Alternative 6
Error	43.7
Cost	21000

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{B \cdot B + C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+82}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{-1}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \]

Alternative 7
Error	44.8
Cost	20816

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+26}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 5.9 \cdot 10^{+85}:\\ \;\;\;\;\left(\sqrt{2} \cdot \frac{-1}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \]

Alternative 8
Error	44.8
Cost	20752

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{+153}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \]

Alternative 9
Error	44.8
Cost	20036

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{C - A \cdot -2}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 10
Error	48.9
Cost	15380

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-268}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-202}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)} \cdot \left(-\sqrt{A + \left(C + A\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 11
Error	49.1
Cost	15120

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -2.15 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-269}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-243}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + A\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 12
Error	45.5
Cost	15044

\[\begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 13
Error	50.5
Cost	14348

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{-139}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(B - \left(C + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-273}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-70}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot t_0\right)\right) \cdot -2}}{t_2}\\ \mathbf{elif}\;B \leq 1.16 \cdot 10^{+27} \lor \neg \left(B \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \end{array} \]

Alternative 14
Error	49.2
Cost	14348

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-70}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 15
Error	50.6
Cost	14233

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_2 := F \cdot \left(t_0 - B \cdot B\right)\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(B - \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-270}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 10^{-260}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A - C\right) - \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot t_0\right)\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+27} \lor \neg \left(B \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \end{array} \]

Alternative 16
Error	50.1
Cost	13576

\[\begin{array}{l} \mathbf{if}\;C \leq -5.5 \cdot 10^{+59}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right) \cdot \left(\left(A - C\right) - \left(C + A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 17
Error	51.8
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C - A\right) - \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(C + A\right)\right) \cdot t_1\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	51.8
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C - A\right) - \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A - C\right) - \left(C + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	51.8
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\\ \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{elif}\;C \leq 1.32 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(C + A\right)\right) \cdot t_1\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	54.5
Cost	8452

\[\begin{array}{l} \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4 - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 21
Error	55.9
Cost	8196

\[\begin{array}{l} \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 22
Error	55.4
Cost	8196

\[\begin{array}{l} \mathbf{if}\;C \leq -2.2 \cdot 10^{+58}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 23
Error	57.3
Cost	7236

\[\begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{-269}:\\ \;\;\;\;-\sqrt{\frac{F}{A - C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - A \cdot -2\right)\right)}}{B}\\ \end{array} \]

Alternative 24
Error	57.8
Cost	6784

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

Reproduce

herbie shell --seed 2023011 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))