ABCF->ab-angle b

Average Error: 52.1 → 40.1

Time: 49.3s

Precision: binary64

Cost: 34188

Math

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

↓

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{F \cdot \left(\left(A + C\right) - t_1\right)}\\ t_3 := \left(A - \left(t_1 - C\right)\right) \cdot \left(F \cdot 2\right)\\ t_4 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;B \leq -4.7 \cdot 10^{+17}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-122}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_4}}{t_4}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{t_3} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{t_0}\right)\right)}{t_4}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{F \cdot \left(A + A\right)} \cdot \left(\sqrt{2 \cdot t_4} \cdot \frac{-1}{t_4}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(F \cdot \left(C + \left(A - t_1\right)\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (hypot B (- A C)))
        (t_2 (sqrt (* F (- (+ A C) t_1))))
        (t_3 (* (- A (- t_1 C)) (* F 2.0)))
        (t_4 (fma B B t_0)))
   (if (<= B -4.7e+17)
     (* t_2 (/ (sqrt 2.0) B))
     (if (<= B -4.7e-122)
       (/ (- (sqrt (* t_3 t_4))) t_4)
       (if (<= B -7.5e-136)
         (/ (* (sqrt t_3) (- (hypot B (sqrt t_0)))) t_4)
         (if (<= B 5.5e-288)
           (sqrt (- (/ F A)))
           (if (<= B 2.25e-129)
             (* (sqrt (* F (+ A A))) (* (sqrt (* 2.0 t_4)) (/ -1.0 t_4)))
             (if (<= B 3.8e-103)
               (/ (- (sqrt (* 2.0 (* t_4 (* F (+ C (- A t_1))))))) t_4)
               (* t_2 (/ (- (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 = A * (C * -4.0);
	double t_1 = hypot(B, (A - C));
	double t_2 = sqrt((F * ((A + C) - t_1)));
	double t_3 = (A - (t_1 - C)) * (F * 2.0);
	double t_4 = fma(B, B, t_0);
	double tmp;
	if (B <= -4.7e+17) {
		tmp = t_2 * (sqrt(2.0) / B);
	} else if (B <= -4.7e-122) {
		tmp = -sqrt((t_3 * t_4)) / t_4;
	} else if (B <= -7.5e-136) {
		tmp = (sqrt(t_3) * -hypot(B, sqrt(t_0))) / t_4;
	} else if (B <= 5.5e-288) {
		tmp = sqrt(-(F / A));
	} else if (B <= 2.25e-129) {
		tmp = sqrt((F * (A + A))) * (sqrt((2.0 * t_4)) * (-1.0 / t_4));
	} else if (B <= 3.8e-103) {
		tmp = -sqrt((2.0 * (t_4 * (F * (C + (A - t_1)))))) / t_4;
	} else {
		tmp = t_2 * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = hypot(B, Float64(A - C))
	t_2 = sqrt(Float64(F * Float64(Float64(A + C) - t_1)))
	t_3 = Float64(Float64(A - Float64(t_1 - C)) * Float64(F * 2.0))
	t_4 = fma(B, B, t_0)
	tmp = 0.0
	if (B <= -4.7e+17)
		tmp = Float64(t_2 * Float64(sqrt(2.0) / B));
	elseif (B <= -4.7e-122)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_4))) / t_4);
	elseif (B <= -7.5e-136)
		tmp = Float64(Float64(sqrt(t_3) * Float64(-hypot(B, sqrt(t_0)))) / t_4);
	elseif (B <= 5.5e-288)
		tmp = sqrt(Float64(-Float64(F / A)));
	elseif (B <= 2.25e-129)
		tmp = Float64(sqrt(Float64(F * Float64(A + A))) * Float64(sqrt(Float64(2.0 * t_4)) * Float64(-1.0 / t_4)));
	elseif (B <= 3.8e-103)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * Float64(F * Float64(C + Float64(A - t_1))))))) / t_4);
	else
		tmp = Float64(t_2 * 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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(A - N[(t$95$1 - C), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(B * B + t$95$0), $MachinePrecision]}, If[LessEqual[B, -4.7e+17], N[(t$95$2 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.7e-122], N[((-N[Sqrt[N[(t$95$3 * t$95$4), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, -7.5e-136], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * (-N[Sqrt[B ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 5.5e-288], N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision], If[LessEqual[B, 2.25e-129], N[(N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.8e-103], N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * N[(F * N[(C + N[(A - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], N[(t$95$2 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.7e17

   1. Initial program 55.0

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

   2. Simplified 53.5

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

      [Start] 55.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 47.7

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

   4. Simplified 47.7

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

      [Start] 47.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.7	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 47.8

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

   6. Taylor expanded in B around -inf 30.9

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

   if -4.7e17 < B < -4.6999999999999999e-122

   1. Initial program 45.5

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

   2. Simplified 39.7

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

      [Start] 45.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -4.6999999999999999e-122 < B < -7.5000000000000003e-136

   1. Initial program 51.1

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

   2. Simplified 45.1

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

      [Start] 51.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 53.1

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

   4. Simplified 51.9

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

      [Start] 53.1	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 51.9	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if -7.5000000000000003e-136 < B < 5.5e-288

   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. Simplified 46.6

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

      [Start] 53.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.3

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

   4. Taylor expanded in B around 0 50.1

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

   5. Simplified 50.1

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

      [Start] 50.1	\[ \sqrt{-1 \cdot \frac{F}{A}} \]
      mul-1-neg [=>] 50.1	\[ \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 5.5e-288 < B < 2.25000000000000015e-129

   1. Initial program 52.6

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

   2. Simplified 46.5

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

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

   3. Applied egg-rr 52.0

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

   4. Simplified 52.0

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

      [Start] 52.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 52.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 52.0	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 53.0

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

   6. Taylor expanded in C around inf 54.9

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

   7. Simplified 54.9

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

      [Start] 54.9	\[ \sqrt{\left(A - -1 \cdot A\right) \cdot F} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      *-commutative [=>] 54.9	\[ \sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      sub-neg [=>] 54.9	\[ \sqrt{F \cdot \color{blue}{\left(A + \left(--1 \cdot A\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      mul-1-neg [=>] 54.9	\[ \sqrt{F \cdot \left(A + \left(-\color{blue}{\left(-A\right)}\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      remove-double-neg [=>] 54.9	\[ \sqrt{F \cdot \left(A + \color{blue}{A}\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]

   if 2.25000000000000015e-129 < B < 3.8000000000000001e-103

   1. Initial program 48.5

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

   2. Simplified 43.4

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

      [Start] 48.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 3.8000000000000001e-103 < B

   1. Initial program 52.1

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

   2. Simplified 49.4

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

      [Start] 52.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.6

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

   4. Simplified 46.6

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

      [Start] 46.6	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.6	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 46.8

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

   6. Taylor expanded in B around inf 35.3

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

   7. Simplified 35.3

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

      [Start] 35.3	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      mul-1-neg [=>] 35.3	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \]
      distribute-neg-frac [=>] 35.3	\[ \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2}}{B}} \]

3. Recombined 7 regimes into one program.

4. Final simplification 40.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.7 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-122}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(F \cdot 2\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{F \cdot \left(A + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	39.2
Cost	34516

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - t_0\right)}\\ t_2 := A - \left(t_0 - C\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := \frac{\sqrt{F \cdot t_2} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_3}\\ \mathbf{if}\;B \leq -6 \cdot 10^{+15}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{\left(t_2 \cdot \left(F \cdot 2\right)\right) \cdot t_3}}{t_3}\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-136}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{+119}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 2
Error	40.6
Cost	28248

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(A + A\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_2}\\ t_4 := \sqrt{F \cdot \left(\left(A + C\right) - t_0\right)}\\ t_5 := \sqrt{2 \cdot t_2}\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{+100}:\\ \;\;\;\;t_4 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.22 \cdot 10^{-114}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{-133}:\\ \;\;\;\;\frac{t_5 \cdot \left(-t_1\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-130}:\\ \;\;\;\;t_1 \cdot \left(t_5 \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-101}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	40.3
Cost	28248

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A + A\right)}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F \cdot \left(\left(A + C\right) - t_1\right)}\\ t_4 := \sqrt{2 \cdot t_2}\\ \mathbf{if}\;B \leq -9 \cdot 10^{+16}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -3.7 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A - \left(t_1 - C\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot t_2}}{t_2}\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{t_4 \cdot \left(-t_0\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-130}:\\ \;\;\;\;t_0 \cdot \left(t_4 \cdot \frac{-1}{t_2}\right)\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C + \left(A - t_1\right)\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	41.3
Cost	27596

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -4.3 \cdot 10^{-132}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{F \cdot \left(A + A\right)} \cdot \left(\sqrt{2 \cdot t_0} \cdot \frac{-1}{t_0}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	41.3
Cost	27532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-132}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.28 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	40.8
Cost	20424

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{-135}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	41.8
Cost	20228

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{-135}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot t_0\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-t_0 \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 8
Error	48.0
Cost	20168

\[\begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 9
Error	48.9
Cost	13832

\[\begin{array}{l} \mathbf{if}\;B \leq -2.95 \cdot 10^{-130}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(\left(A + C\right) - B\right)}\right)\\ \end{array} \]

Alternative 10
Error	48.9
Cost	13704

\[\begin{array}{l} \mathbf{if}\;B \leq -5.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 11
Error	48.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;B \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 12
Error	53.6
Cost	13584

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{if}\;A \leq -3600000:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -1.4 \cdot 10^{-273}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 2 \cdot 10^{-170}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C - B\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{A}}\\ \end{array} \]

Alternative 13
Error	54.4
Cost	8848

\[\begin{array}{l} t_0 := -\frac{\sqrt{\left(\left(A + \left(C - B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ t_1 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -2.65 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 6.5 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 7 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 14
Error	53.6
Cost	8584

\[\begin{array}{l} t_0 := F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-135}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(B + C\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(\left(A + \left(C - B\right)\right) \cdot t_0\right) \cdot -2}}{t_1}\\ \end{array} \]

Alternative 15
Error	53.6
Cost	7053

\[\begin{array}{l} \mathbf{if}\;C \leq -2.9 \cdot 10^{+151} \lor \neg \left(C \leq -7.2 \cdot 10^{-218}\right) \land C \leq 2.3 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 16
Error	56.3
Cost	6656

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

Alternative 17
Error	63.3
Cost	6592

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

Reproduce

herbie shell --seed 2023059 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))