ABCF->ab-angle a

Average Error: 52.4 → 31.1

Time: 1.2min

Precision: binary64

Cost: 34384

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := C + \left(A + t_1\right)\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -9.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{t_2}}{\frac{-\mathsf{fma}\left(B, B, t_3\right)}{\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_3}}}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_3 + B \cdot B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_2} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0))))
        (t_1 (hypot B (- A C)))
        (t_2 (+ C (+ A t_1)))
        (t_3 (* -4.0 (* A C))))
   (if (<= B -9.6e+55)
     (/ (* (sqrt (+ A (+ C t_1))) (sqrt (* F 2.0))) B)
     (if (<= B 8e-280)
       (/
        (sqrt t_2)
        (/
         (- (fma B B t_3))
         (sqrt (+ (* (* F 2.0) (* B B)) (* (* F 2.0) t_3)))))
       (if (<= B 3.1e-104)
         (/ (- (pow (* (* A (* F (* A C))) -16.0) 0.5)) (+ t_3 (* B B)))
         (if (<= B 1.75e+82)
           (/ (* (sqrt (* F t_2)) (- (sqrt (* 2.0 t_0)))) t_0)
           (* (* (sqrt (+ C (hypot C B))) (sqrt F)) (/ (- (sqrt 2.0)) B))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = hypot(B, (A - C));
	double t_2 = C + (A + t_1);
	double t_3 = -4.0 * (A * C);
	double tmp;
	if (B <= -9.6e+55) {
		tmp = (sqrt((A + (C + t_1))) * sqrt((F * 2.0))) / B;
	} else if (B <= 8e-280) {
		tmp = sqrt(t_2) / (-fma(B, B, t_3) / sqrt((((F * 2.0) * (B * B)) + ((F * 2.0) * t_3))));
	} else if (B <= 3.1e-104) {
		tmp = -pow(((A * (F * (A * C))) * -16.0), 0.5) / (t_3 + (B * B));
	} else if (B <= 1.75e+82) {
		tmp = (sqrt((F * t_2)) * -sqrt((2.0 * t_0))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = hypot(B, Float64(A - C))
	t_2 = Float64(C + Float64(A + t_1))
	t_3 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if (B <= -9.6e+55)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + t_1))) * sqrt(Float64(F * 2.0))) / B);
	elseif (B <= 8e-280)
		tmp = Float64(sqrt(t_2) / Float64(Float64(-fma(B, B, t_3)) / sqrt(Float64(Float64(Float64(F * 2.0) * Float64(B * B)) + Float64(Float64(F * 2.0) * t_3)))));
	elseif (B <= 3.1e-104)
		tmp = Float64(Float64(-(Float64(Float64(A * Float64(F * Float64(A * C))) * -16.0) ^ 0.5)) / Float64(t_3 + Float64(B * B)));
	elseif (B <= 1.75e+82)
		tmp = Float64(Float64(sqrt(Float64(F * t_2)) * Float64(-sqrt(Float64(2.0 * t_0)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(C + N[(A + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -9.6e+55], N[(N[(N[Sqrt[N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 8e-280], N[(N[Sqrt[t$95$2], $MachinePrecision] / N[((-N[(B * B + t$95$3), $MachinePrecision]) / N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] + N[(N[(F * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.1e-104], N[((-N[Power[N[(N[(A * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision], 0.5], $MachinePrecision]) / N[(t$95$3 + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.75e+82], N[(N[(N[Sqrt[N[(F * t$95$2), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -9.5999999999999997e55

   1. Initial program 57.8

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

   2. Simplified 57.8

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

      [Start] 57.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 52.9

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

   4. Simplified 52.9

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

      [Start] 52.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 52.9	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 52.9	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 52.9	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 52.9

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

   6. Simplified 52.9

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

      [Start] 52.9	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r* [=>] 52.9	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      distribute-rgt-neg-out [=>] 52.9	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      distribute-lft-neg-out [<=] 52.9	\[ \color{blue}{\left(\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [<=] 52.9	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      +-commutative [=>] 52.9	\[ \left(-\sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 52.9	\[ \left(-\sqrt{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 52.9	\[ \left(-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Taylor expanded in B around -inf 23.0

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

   8. Simplified 23.0

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

      [Start] 23.0	\[ \left(-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}\right) \cdot \left(-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\right) \]
      mul-1-neg [=>] 23.0	\[ \left(-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}\right) \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)} \]
      *-commutative [=>] 23.0	\[ \left(-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}\right) \cdot \left(-\color{blue}{\sqrt{F} \cdot \frac{\sqrt{2}}{B}}\right) \]

   9. Applied egg-rr 15.1

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

   if -9.5999999999999997e55 < B < 7.9999999999999997e-280

   1. Initial program 49.0

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

   2. Simplified 49.0

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

      [Start] 49.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 38.7

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

   4. Simplified 38.8

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

      [Start] 38.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 38.7	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 38.8	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 38.8	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 38.9

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

   6. Simplified 38.7

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

      [Start] 38.9	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r* [=>] 38.9	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      distribute-rgt-neg-out [=>] 38.9	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      distribute-lft-neg-out [<=] 38.9	\[ \color{blue}{\left(\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [<=] 38.9	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      +-commutative [=>] 38.9	\[ \left(-\sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 38.8	\[ \left(-\sqrt{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}\right) \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 38.7	\[ \left(-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Applied egg-rr 38.8

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

   8. Simplified 38.7

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

      [Start] 38.8	\[ \frac{\sqrt{F \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]
      *-commutative [=>] 38.8	\[ \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]
      associate-/l* [=>] 38.8	\[ \color{blue}{\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{F \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}}}} \]
      +-commutative [=>] 38.8	\[ \frac{\sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{F \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}}} \]
      associate-+l+ [=>] 38.7	\[ \frac{\sqrt{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{F \cdot \left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right)}}} \]
      *-commutative [=>] 38.7	\[ \frac{\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\color{blue}{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot 2\right) \cdot F}}}} \]
      associate-*l* [=>] 38.7	\[ \frac{\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}}} \]
      *-commutative [=>] 38.7	\[ \frac{\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(F \cdot 2\right)}}}} \]

   9. Applied egg-rr 38.7

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

   if 7.9999999999999997e-280 < B < 3.09999999999999976e-104

   1. Initial program 52.5

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

   2. Simplified 52.5

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

      [Start] 52.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 53.9

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

   4. Simplified 53.9

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

      [Start] 53.9	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 53.9	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 53.9	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 48.8

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

   if 3.09999999999999976e-104 < B < 1.75e82

   1. Initial program 43.0

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

   2. Simplified 37.7

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

      [Start] 43.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 37.7

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

   4. Simplified 37.7

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

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

   if 1.75e82 < B

   1. Initial program 60.3

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

   2. Simplified 59.3

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

      [Start] 60.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 56.9

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

   4. Simplified 56.9

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

      [Start] 56.9	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 56.9	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 56.9	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 56.9	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      *-commutative [=>] 56.9	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]
      +-commutative [=>] 56.9	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{{C}^{2} + {B}^{2}}}\right)} \]
      unpow2 [=>] 56.9	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{C \cdot C} + {B}^{2}}\right)} \]
      unpow2 [=>] 56.9	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{C \cdot C + \color{blue}{B \cdot B}}\right)} \]

   5. Applied egg-rr 17.0

      \[\leadsto \frac{-\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 31.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}}}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	30.0
Cost	34384

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{C + \left(A + t_1\right)}\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_2}{\frac{-\mathsf{fma}\left(B, B, t_3\right)}{\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_3}}}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_3 + B \cdot B}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+86}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	29.9
Cost	28368

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{\sqrt{C + \left(A + t_0\right)}}{\frac{-\mathsf{fma}\left(B, B, t_1\right)}{\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_1}}}\\ \mathbf{if}\;B \leq -6.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_1 + B \cdot B}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	30.1
Cost	28112

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-t_1\right)}{t_0}\\ \mathbf{if}\;B \leq -8.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{t_1 \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 7.4 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	32.3
Cost	27592

\[\begin{array}{l} t_0 := \frac{-\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{C + \left(A + t_1\right)}\\ \mathbf{if}\;B \leq -8.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-280}:\\ \;\;\;\;\left(-t_2\right) \cdot \frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{+191}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot t_0\\ \end{array} \]

Alternative 5
Error	32.3
Cost	27592

\[\begin{array}{l} t_0 := F \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2}}{B}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := C + \left(A + t_2\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_2\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_3} \cdot \left(-\sqrt{-4 \cdot t_0}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_0\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{t_3} \cdot \left(\sqrt{F} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot t_1\\ \end{array} \]

Alternative 6
Error	32.3
Cost	27464

\[\begin{array}{l} t_0 := F \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2}}{B}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{C + \left(A + t_2\right)}\\ \mathbf{if}\;B \leq -3 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_2\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 2.95 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{t_0 \cdot -8}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4.45 \cdot 10^{-104}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_0\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{+191}:\\ \;\;\;\;t_3 \cdot \left(\sqrt{F} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot t_1\\ \end{array} \]

Alternative 7
Error	32.3
Cost	27088

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{2}}{B}\\ t_2 := F \cdot \left(A \cdot C\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \sqrt{A + \left(C + t_3\right)}\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{t_4 \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot -8} \cdot \left(-t_4\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-110}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_2\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+189}:\\ \;\;\;\;\sqrt{C + \left(A + t_3\right)} \cdot \left(\sqrt{F} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot t_1\\ \end{array} \]

Alternative 8
Error	32.4
Cost	26832

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_3 := F \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{t_2 \cdot t_1}{B}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot -8} \cdot \left(-t_2\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.16 \cdot 10^{-116}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_3\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+192}:\\ \;\;\;\;t_2 \cdot \frac{-t_1}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	34.5
Cost	21192

\[\begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \sqrt{t_1}\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{t_2 \cdot t_0}{B}\\ \mathbf{elif}\;B \leq 2.95 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_3\right) \cdot \left(t_1 \cdot 2\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.38 \cdot 10^{-108}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_3 + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-t_0}{B}\\ \end{array} \]

Alternative 10
Error	33.7
Cost	21192

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_3 := F \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{t_2 \cdot t_1}{B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot -8} \cdot \left(-t_2\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot t_3\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-t_1}{B}\\ \end{array} \]

Alternative 11
Error	34.4
Cost	21000

\[\begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.12 \cdot 10^{-106}:\\ \;\;\;\;\frac{t_1 \cdot t_0}{B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-198}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t_0}{B}\\ \end{array} \]

Alternative 12
Error	42.6
Cost	20696

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := -\sqrt{2}\\ t_2 := \frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{if}\;B \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;\left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{-B}\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(F \cdot 2\right) \cdot \left(B \cdot B\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{t_1}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 13
Error	35.1
Cost	20424

\[\begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{-36}:\\ \;\;\;\;\frac{t_1 \cdot t_0}{B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t_0}{B}\\ \end{array} \]

Alternative 14
Error	39.6
Cost	20300

\[\begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{B} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-103}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 15
Error	38.4
Cost	20300

\[\begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F \cdot 2}}{B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 16
Error	42.9
Cost	19844

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{\left(\left(F \cdot 2\right) \cdot \left(B \cdot B\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{+63}:\\ \;\;\;\;\left(\sqrt{F} \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{-B}\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 7.1 \cdot 10^{-234}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 17
Error	48.5
Cost	15188

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{\left(\left(F \cdot 2\right) \cdot \left(B \cdot B\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.35 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 18
Error	51.8
Cost	15064

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 + B \cdot B\\ t_2 := F \cdot t_1\\ t_3 := \frac{-\sqrt{\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -450000000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -4 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq -2.9 \cdot 10^{-261}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{elif}\;A \leq 6.2 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 3.8 \cdot 10^{+69}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 19
Error	51.7
Cost	14344

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ t_2 := t_0 + B \cdot B\\ t_3 := F \cdot t_2\\ \mathbf{if}\;A \leq -4300000000000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -1.52 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;A \leq -1.06 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.05 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq 2.22 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t_2}\\ \mathbf{elif}\;A \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 20
Error	52.0
Cost	14240

\[\begin{array}{l} t_0 := A \cdot \left(A \cdot C\right)\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_2 := F \cdot t_1\\ t_3 := \left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{if}\;A \leq -220000000:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -8.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{t_0}{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}} \cdot \left(F \cdot 16\right)}{t_1}\\ \mathbf{elif}\;A \leq -7.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -5.4 \cdot 10^{-260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 5.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 10^{-109}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot -16\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{+70}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 21
Error	51.7
Cost	14240

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ t_3 := A \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -240000000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{t_3}{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}} \cdot \left(F \cdot 16\right)}{t_0}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 2.1 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	53.8
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := -\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -1880000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -2.75 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 9.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A + \left(\frac{-0.5}{\frac{B}{A \cdot A}} - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	51.2
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -6.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;C \leq 2.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Error	51.1
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.75 \cdot 10^{-278}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;C \leq 5 \cdot 10^{-177}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 8.2 \cdot 10^{+64}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 25
Error	54.0
Cost	8336

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := -\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -64000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -9 \cdot 10^{-60}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{elif}\;A \leq -3.15 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 2.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \end{array} \]

Alternative 26
Error	53.7
Cost	7940

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;C \leq 1.25 \cdot 10^{-104}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 27
Error	56.2
Cost	7876

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq 1.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{t_0}\\ \end{array} \]

Alternative 28
Error	57.5
Cost	7812

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq 5.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{t_0}\\ \end{array} \]

Alternative 29
Error	57.1
Cost	7812

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq 1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{t_0}\\ \end{array} \]

Alternative 30
Error	62.3
Cost	7488

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

Alternative 31
Error	63.1
Cost	6976

\[\sqrt{\frac{F}{A}} \cdot \left(0.5 \cdot \frac{B}{C}\right) \]

Alternative 32
Error	63.1
Cost	6976

\[\sqrt{\frac{F}{A}} \cdot \frac{0.5}{\frac{C}{B}} \]

Alternative 33
Error	63.1
Cost	6976

\[\frac{0.5}{\frac{\frac{C}{B}}{\sqrt{\frac{F}{A}}}} \]

Reproduce

herbie shell --seed 2023060 
(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))))