ABCF->ab-angle a

Average Error: 52.7 → 33.3

Time: 1.3min

Precision: binary64

Cost: 34712

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\\ t_3 := t_0 \cdot \frac{-1}{t_2}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-10}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{t_2}\right) \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_4\right) \cdot \left(C + C\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right) \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot t_2\right)} \cdot t_3\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2 - \frac{{B}^{2}}{A} \cdot 0.5\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 10^{+150}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)\right) \cdot t_3\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

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 (sqrt (+ C (+ A (hypot (- A C) B)))))
        (t_1 (/ (sqrt 2.0) B))
        (t_2 (fma B B (* -4.0 (* C A))))
        (t_3 (* t_0 (/ -1.0 t_2)))
        (t_4 (fma B B (* A (* C -4.0)))))
   (if (<= B -1e+154)
     (* (sqrt (* F (+ C (hypot C B)))) t_1)
     (if (<= B -4.3e-10)
       (* (* t_0 (/ 1.0 t_2)) (* (sqrt 2.0) (* B (sqrt F))))
       (if (<= B -7e-159)
         (- (/ (sqrt (* 2.0 (* (* F t_4) (+ C C)))) t_4))
         (if (<= B -6.2e-246)
           (sqrt (- (/ F A)))
           (if (<= B 1.45e-127)
             (*
              (sqrt (* 2.0 (- (* F (* B B)) (* F (* (* C A) 4.0)))))
              (* (* (/ (sqrt 2.0) A) (sqrt (/ 1.0 C))) 0.25))
             (if (<= B 3.3e-35)
               (* (sqrt (* 2.0 (* F t_2))) t_3)
               (if (<= B 5.5e+35)
                 (/
                  (-
                   (sqrt
                    (*
                     t_4
                     (* 2.0 (* F (- (* C 2.0) (* (/ (pow B 2.0) A) 0.5)))))))
                  t_4)
                 (if (<= B 1e+150)
                   (* (* (sqrt F) (* B (sqrt 2.0))) t_3)
                   (if (<= B 9e+230)
                     (* t_1 (- (sqrt (* F (+ C (hypot B C))))))
                     (* (sqrt (/ F B)) (- (sqrt 2.0))))))))))))))

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 = sqrt((C + (A + hypot((A - C), B))));
	double t_1 = sqrt(2.0) / B;
	double t_2 = fma(B, B, (-4.0 * (C * A)));
	double t_3 = t_0 * (-1.0 / t_2);
	double t_4 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -1e+154) {
		tmp = sqrt((F * (C + hypot(C, B)))) * t_1;
	} else if (B <= -4.3e-10) {
		tmp = (t_0 * (1.0 / t_2)) * (sqrt(2.0) * (B * sqrt(F)));
	} else if (B <= -7e-159) {
		tmp = -(sqrt((2.0 * ((F * t_4) * (C + C)))) / t_4);
	} else if (B <= -6.2e-246) {
		tmp = sqrt(-(F / A));
	} else if (B <= 1.45e-127) {
		tmp = sqrt((2.0 * ((F * (B * B)) - (F * ((C * A) * 4.0))))) * (((sqrt(2.0) / A) * sqrt((1.0 / C))) * 0.25);
	} else if (B <= 3.3e-35) {
		tmp = sqrt((2.0 * (F * t_2))) * t_3;
	} else if (B <= 5.5e+35) {
		tmp = -sqrt((t_4 * (2.0 * (F * ((C * 2.0) - ((pow(B, 2.0) / A) * 0.5)))))) / t_4;
	} else if (B <= 1e+150) {
		tmp = (sqrt(F) * (B * sqrt(2.0))) * t_3;
	} else if (B <= 9e+230) {
		tmp = t_1 * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	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 = sqrt(Float64(C + Float64(A + hypot(Float64(A - C), B))))
	t_1 = Float64(sqrt(2.0) / B)
	t_2 = fma(B, B, Float64(-4.0 * Float64(C * A)))
	t_3 = Float64(t_0 * Float64(-1.0 / t_2))
	t_4 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -1e+154)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * t_1);
	elseif (B <= -4.3e-10)
		tmp = Float64(Float64(t_0 * Float64(1.0 / t_2)) * Float64(sqrt(2.0) * Float64(B * sqrt(F))));
	elseif (B <= -7e-159)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_4) * Float64(C + C)))) / t_4));
	elseif (B <= -6.2e-246)
		tmp = sqrt(Float64(-Float64(F / A)));
	elseif (B <= 1.45e-127)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * Float64(B * B)) - Float64(F * Float64(Float64(C * A) * 4.0))))) * Float64(Float64(Float64(sqrt(2.0) / A) * sqrt(Float64(1.0 / C))) * 0.25));
	elseif (B <= 3.3e-35)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * t_2))) * t_3);
	elseif (B <= 5.5e+35)
		tmp = Float64(Float64(-sqrt(Float64(t_4 * Float64(2.0 * Float64(F * Float64(Float64(C * 2.0) - Float64(Float64((B ^ 2.0) / A) * 0.5))))))) / t_4);
	elseif (B <= 1e+150)
		tmp = Float64(Float64(sqrt(F) * Float64(B * sqrt(2.0))) * t_3);
	elseif (B <= 9e+230)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(C + N[(A + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1e+154], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[B, -4.3e-10], N[(N[(t$95$0 * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(B * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7e-159], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$4), $MachinePrecision] * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), If[LessEqual[B, -6.2e-246], N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision], If[LessEqual[B, 1.45e-127], N[(N[Sqrt[N[(2.0 * N[(N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(F * N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] / A), $MachinePrecision] * N[Sqrt[N[(1.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.3e-35], N[(N[Sqrt[N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[B, 5.5e+35], N[((-N[Sqrt[N[(t$95$4 * N[(2.0 * N[(F * N[(N[(C * 2.0), $MachinePrecision] - N[(N[(N[Power[B, 2.0], $MachinePrecision] / A), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 1e+150], N[(N[(N[Sqrt[F], $MachinePrecision] * N[(B * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[B, 9e+230], N[(t$95$1 * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 10 regimes

2. if B < -1.00000000000000004e154

   1. Initial program 64.0

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

   2. Applied egg-rr 64.0

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

   3. Taylor expanded in A around 0 64.0

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

   4. Simplified 31.3

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

   if -1.00000000000000004e154 < B < -4.30000000000000014e-10

   1. Initial program 45.0

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

   2. Applied egg-rr 33.0

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

   3. Applied egg-rr 33.0

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

   4. Applied egg-rr 33.0

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

   5. Taylor expanded in B around -inf 26.7

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

   6. Simplified 26.7

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

   if -4.30000000000000014e-10 < B < -7.00000000000000005e-159

   1. Initial program 47.4

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

   2. Simplified 41.0

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

   3. Taylor expanded in A around -inf 36.6

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

   if -7.00000000000000005e-159 < B < -6.2000000000000001e-246

   1. Initial program 53.1

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

   2. Applied egg-rr 58.3

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

   3. Taylor expanded in A around -inf 39.8

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]

   4. Simplified 39.8

      \[\leadsto \sqrt{\color{blue}{\frac{-F}{A}}} \]

   if -6.2000000000000001e-246 < B < 1.45e-127

   1. Initial program 52.8

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

   2. Applied egg-rr 42.6

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

   3. Applied egg-rr 42.8

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

   4. Applied egg-rr 42.8

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

   5. Taylor expanded in A around -inf 36.0

      \[\leadsto \left(-\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot F + \left(-4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}\right) \cdot \color{blue}{\left(-0.25 \cdot \left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right)\right)} \]

   if 1.45e-127 < B < 3.3e-35

   1. Initial program 47.9

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

   2. Applied egg-rr 36.9

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

   3. Applied egg-rr 36.9

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

   if 3.3e-35 < B < 5.50000000000000001e35

   1. Initial program 43.4

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

   2. Simplified 38.1

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

   3. Applied egg-rr 39.1

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

   4. Taylor expanded in A around -inf 44.3

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

   if 5.50000000000000001e35 < B < 9.99999999999999981e149

   1. Initial program 48.9

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

   2. Applied egg-rr 36.3

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

   3. Applied egg-rr 36.4

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

   4. Taylor expanded in B around inf 27.5

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

   if 9.99999999999999981e149 < B < 8.9999999999999998e230

   1. Initial program 63.6

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

   2. Simplified 63.6

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

   3. Applied egg-rr 63.6

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

   4. Taylor expanded in A around 0 63.6

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

   5. Simplified 63.6

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

   6. Taylor expanded in A around 0 62.3

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

   7. Simplified 28.2

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

   if 8.9999999999999998e230 < B

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Applied egg-rr 64.0

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

   4. Taylor expanded in A around 0 64.0

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

   5. Simplified 64.0

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

   6. Taylor expanded in C around 0 29.6

      \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 10 regimes into one program.

4. Final simplification 33.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-10}:\\ \;\;\;\;\left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right) \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(C + C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right) \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2 - \frac{{B}^{2}}{A} \cdot 0.5\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 10^{+150}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	34.8
Cost	34592

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \mathsf{hypot}\left(A - C, B\right)\\ t_2 := \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_4\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(C + C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \left(F \cdot t_2\right)}\right) \cdot \sqrt{C + C}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)}\right) \cdot \frac{\sqrt{A + \left(C + t_1\right)}}{t_2}\\ \mathbf{elif}\;B \leq 10^{+136}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)\right) \cdot \left(\sqrt{C + \left(A + t_1\right)} \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_4 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 2
Error	34.0
Cost	33800

\[\begin{array}{l} t_0 := \sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)}\\ t_1 := \sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-10}:\\ \;\;\;\;\left(t_0 \cdot \frac{1}{t_3}\right) \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_4\right) \cdot \left(C + C\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-127}:\\ \;\;\;\;t_1 \cdot \left(\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right) \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{-1}{t_3}\right)\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2 - \frac{{B}^{2}}{A} \cdot 0.5\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 3
Error	35.0
Cost	28636

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := C + \mathsf{hypot}\left(C, B\right)\\ t_4 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot t_3} \cdot t_4\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \left(F \cdot t_1\right)}\right) \cdot \sqrt{C + C}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-62}:\\ \;\;\;\;\left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)}\right) \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(B \cdot \sqrt{F}\right) \cdot \left(-\sqrt{t_3 \cdot 2}\right)}{t_2}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_4 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 4
Error	35.1
Cost	28508

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}\right) \cdot \sqrt{C + C}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 0.015:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2 - \frac{{B}^{2}}{A} \cdot 0.5\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 5
Error	35.4
Cost	28124

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{-\frac{F}{A}}\\ t_3 := \frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}\right) \cdot \sqrt{C + C}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6
Error	35.1
Cost	22104

\[\begin{array}{l} t_0 := F \cdot \left(B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ t_3 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_3\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_0 - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right) \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(A \cdot \left(C \cdot 4\right)\right) - t_0\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	35.2
Cost	21660

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ t_3 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_3\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(\frac{\sqrt{2} \cdot \sqrt{\frac{1}{C}}}{A} \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8
Error	35.2
Cost	21660

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ t_3 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_3\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-267}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right) - F \cdot \left(\left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(\left(\frac{\sqrt{2}}{A} \cdot \sqrt{\frac{1}{C}}\right) \cdot 0.25\right)\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.55 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9
Error	34.7
Cost	21264

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{t_0 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 0.0155:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10
Error	34.6
Cost	21264

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.36 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-159}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 0.0155:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11
Error	37.4
Cost	20300

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -9 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot t_0\\ \mathbf{elif}\;B \leq 0.015:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+230}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12
Error	38.3
Cost	19972

\[\begin{array}{l} \mathbf{if}\;B \leq -9 \cdot 10^{-57}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 0.015:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13
Error	44.0
Cost	14348

\[\begin{array}{l} \mathbf{if}\;C \leq -1.8 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;C \leq 253.58902972973797:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq 7.235174397181661 \cdot 10^{+134}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 14
Error	45.9
Cost	13448

\[\begin{array}{l} \mathbf{if}\;A \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq 2.3 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 15
Error	48.8
Cost	13188

\[\begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{F \cdot C} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 16
Error	50.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot C} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 17
Error	63.1
Cost	6656

\[{\left(\frac{F}{C}\right)}^{0.5} \]

Alternative 18
Error	51.3
Cost	6656

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

Reproduce

herbie shell --seed 2022283 
(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))))