ABCF->ab-angle b

Average Error: 52.2 → 42.5

Time: 50.0s

Precision: binary64

Cost: 47448

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_2 := A \cdot \left(C \cdot -4\right)\\ t_3 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_4 := -\sqrt{2 \cdot t_1}\\ t_5 := \log \left(F \cdot -2\right)\\ t_6 := \sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)} \cdot \frac{t_4}{t_1}\\ t_7 := \mathsf{fma}\left(B, B, t_2\right)\\ \mathbf{if}\;B \leq -7 \cdot 10^{-19}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-45}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-136}:\\ \;\;\;\;\frac{t_4 \cdot {\left(e^{0.25 \cdot \left(t_5 - \log \left(\frac{-1}{C}\right)\right)}\right)}^{2}}{t_7}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(A + \left(C - t_0\right)\right) \cdot \left(F \cdot 2\right)\right)}}{t_7}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{t_4 \cdot {\left(e^{0.25 \cdot \left(t_5 - \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{t_7}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-30}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (fma A (* C -4.0) (* B B)))
        (t_2 (* A (* C -4.0)))
        (t_3 (sqrt (* F (+ C (- A t_0)))))
        (t_4 (- (sqrt (* 2.0 t_1))))
        (t_5 (log (* F -2.0)))
        (t_6
         (*
          (sqrt (* F (- C (fma -1.0 C (* 0.5 (/ (* B B) (- A C)))))))
          (/ t_4 t_1)))
        (t_7 (fma B B t_2)))
   (if (<= B -7e-19)
     (* t_3 (/ (sqrt 2.0) B))
     (if (<= B -4.4e-35)
       (sqrt (/ (- F) A))
       (if (<= B -3.3e-45)
         t_6
         (if (<= B -1e-136)
           (/ (* t_4 (pow (exp (* 0.25 (- t_5 (log (/ -1.0 C))))) 2.0)) t_7)
           (if (<= B -5e-164)
             (/ (- (sqrt (* t_2 (* (+ A (- C t_0)) (* F 2.0))))) t_7)
             (if (<= B 1.5e-203)
               (/
                (* t_4 (pow (exp (* 0.25 (- t_5 (log (/ -1.0 A))))) 2.0))
                t_7)
               (if (<= B 8e-30) t_6 (* t_3 (/ (- (sqrt 2.0)) B)))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = fma(A, (C * -4.0), (B * B));
	double t_2 = A * (C * -4.0);
	double t_3 = sqrt((F * (C + (A - t_0))));
	double t_4 = -sqrt((2.0 * t_1));
	double t_5 = log((F * -2.0));
	double t_6 = sqrt((F * (C - fma(-1.0, C, (0.5 * ((B * B) / (A - C))))))) * (t_4 / t_1);
	double t_7 = fma(B, B, t_2);
	double tmp;
	if (B <= -7e-19) {
		tmp = t_3 * (sqrt(2.0) / B);
	} else if (B <= -4.4e-35) {
		tmp = sqrt((-F / A));
	} else if (B <= -3.3e-45) {
		tmp = t_6;
	} else if (B <= -1e-136) {
		tmp = (t_4 * pow(exp((0.25 * (t_5 - log((-1.0 / C))))), 2.0)) / t_7;
	} else if (B <= -5e-164) {
		tmp = -sqrt((t_2 * ((A + (C - t_0)) * (F * 2.0)))) / t_7;
	} else if (B <= 1.5e-203) {
		tmp = (t_4 * pow(exp((0.25 * (t_5 - log((-1.0 / A))))), 2.0)) / t_7;
	} else if (B <= 8e-30) {
		tmp = t_6;
	} else {
		tmp = t_3 * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_2 = Float64(A * Float64(C * -4.0))
	t_3 = sqrt(Float64(F * Float64(C + Float64(A - t_0))))
	t_4 = Float64(-sqrt(Float64(2.0 * t_1)))
	t_5 = log(Float64(F * -2.0))
	t_6 = Float64(sqrt(Float64(F * Float64(C - fma(-1.0, C, Float64(0.5 * Float64(Float64(B * B) / Float64(A - C))))))) * Float64(t_4 / t_1))
	t_7 = fma(B, B, t_2)
	tmp = 0.0
	if (B <= -7e-19)
		tmp = Float64(t_3 * Float64(sqrt(2.0) / B));
	elseif (B <= -4.4e-35)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (B <= -3.3e-45)
		tmp = t_6;
	elseif (B <= -1e-136)
		tmp = Float64(Float64(t_4 * (exp(Float64(0.25 * Float64(t_5 - log(Float64(-1.0 / C))))) ^ 2.0)) / t_7);
	elseif (B <= -5e-164)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(Float64(A + Float64(C - t_0)) * Float64(F * 2.0))))) / t_7);
	elseif (B <= 1.5e-203)
		tmp = Float64(Float64(t_4 * (exp(Float64(0.25 * Float64(t_5 - log(Float64(-1.0 / A))))) ^ 2.0)) / t_7);
	elseif (B <= 8e-30)
		tmp = t_6;
	else
		tmp = Float64(t_3 * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(F * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = (-N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$5 = N[Log[N[(F * -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(F * N[(C - N[(-1.0 * C + N[(0.5 * N[(N[(B * B), $MachinePrecision] / N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(B * B + t$95$2), $MachinePrecision]}, If[LessEqual[B, -7e-19], N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.4e-35], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, -3.3e-45], t$95$6, If[LessEqual[B, -1e-136], N[(N[(t$95$4 * N[Power[N[Exp[N[(0.25 * N[(t$95$5 - N[Log[N[(-1.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision], If[LessEqual[B, -5e-164], N[((-N[Sqrt[N[(t$95$2 * N[(N[(A + N[(C - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$7), $MachinePrecision], If[LessEqual[B, 1.5e-203], N[(N[(t$95$4 * N[Power[N[Exp[N[(0.25 * N[(t$95$5 - N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision], If[LessEqual[B, 8e-30], t$95$6, N[(t$95$3 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -7.00000000000000031e-19

   1. Initial program 54.1

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

   2. Simplified 52.0

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

      [Start] 54.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 47.9

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

   4. Simplified 47.9

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

      [Start] 47.9	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.9	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 47.9	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 47.9

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

   6. Simplified 47.9

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

      [Start] 47.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-lft-neg-out [=>] 47.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)} \]
      associate-*r/ [=>] 47.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot 1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]
      *-rgt-identity [=>] 47.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-frac-neg [<=] 47.9	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around -inf 33.4

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

   if -7.00000000000000031e-19 < B < -4.39999999999999987e-35

   1. Initial program 34.3

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

   2. Simplified 30.3

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

      [Start] 34.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 30.4

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

   4. Simplified 30.4

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

      [Start] 30.4	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 30.4	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 30.4	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 62.0

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

   6. Simplified 61.4

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

      [Start] 62.0	\[ \sqrt{\frac{F \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}}} \]
      associate-/l* [=>] 61.4	\[ \sqrt{\color{blue}{\frac{F}{\frac{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}}{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}}} \]
      associate-/r/ [=>] 61.5	\[ \sqrt{\color{blue}{\frac{F}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}} \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}} \]
      *-commutative [=>] 61.5	\[ \sqrt{\frac{F}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}} \cdot \color{blue}{\left(\left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}} \]
      associate--r- [=>] 61.4	\[ \sqrt{\frac{F}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}} \cdot \left(\left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}\right)} \]
      +-commutative [=>] 61.4	\[ \sqrt{\frac{F}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{2}} \cdot \left(\left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)} \]

   7. Taylor expanded in C around -inf 58.6

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

   8. Simplified 58.6

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

      [Start] 58.6	\[ \sqrt{-1 \cdot \frac{F}{A}} \]
      associate-*r/ [=>] 58.6	\[ \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]
      neg-mul-1 [<=] 58.6	\[ \sqrt{\frac{\color{blue}{-F}}{A}} \]

   if -4.39999999999999987e-35 < B < -3.3000000000000001e-45 or 1.5000000000000001e-203 < B < 8.000000000000001e-30

   1. Initial program 48.1

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

   2. Simplified 43.0

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

      [Start] 48.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.5

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

   4. Simplified 46.5

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

      [Start] 46.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 46.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 46.8

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

   6. Simplified 46.5

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

      [Start] 46.8	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-lft-neg-out [=>] 46.8	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)} \]
      associate-*r/ [=>] 46.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot 1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]
      *-rgt-identity [=>] 46.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-frac-neg [<=] 46.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around 0 53.2

      \[\leadsto \sqrt{F \cdot \left(C - \color{blue}{\left(-1 \cdot C + 0.5 \cdot \frac{{B}^{2}}{A - C}\right)}\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   8. Simplified 53.2

      \[\leadsto \sqrt{F \cdot \left(C - \color{blue}{\mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)}\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      Proof

      [Start] 53.2	\[ \sqrt{F \cdot \left(C - \left(-1 \cdot C + 0.5 \cdot \frac{{B}^{2}}{A - C}\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      fma-def [=>] 53.2	\[ \sqrt{F \cdot \left(C - \color{blue}{\mathsf{fma}\left(-1, C, 0.5 \cdot \frac{{B}^{2}}{A - C}\right)}\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 53.2	\[ \sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{\color{blue}{B \cdot B}}{A - C}\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if -3.3000000000000001e-45 < B < -1e-136

   1. Initial program 48.0

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

   2. Simplified 42.2

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

      [Start] 48.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 42.5

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

   4. Simplified 42.5

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

      [Start] 42.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 42.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 42.5	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 42.5

      \[\leadsto \frac{-\color{blue}{{\left({\left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}^{0.25}\right)}^{2}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   6. Taylor expanded in C around -inf 50.2

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

   if -1e-136 < B < -4.99999999999999962e-164

   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. Simplified 47.4

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

      [Start] 52.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 49.6

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

   4. Simplified 49.6

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

      [Start] 49.6	\[ \frac{-\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.6	\[ \frac{-\sqrt{\color{blue}{\left(\left(A \cdot C\right) \cdot -4\right)} \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [<=] 49.6	\[ \frac{-\sqrt{\color{blue}{\left(A \cdot \left(C \cdot -4\right)\right)} \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if -4.99999999999999962e-164 < B < 1.5000000000000001e-203

   1. Initial program 53.7

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

   2. Simplified 47.8

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

      [Start] 53.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 54.2

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

   4. Simplified 54.2

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

      [Start] 54.2	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 54.2	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 54.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 54.3

      \[\leadsto \frac{-\color{blue}{{\left({\left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}^{0.25}\right)}^{2}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   6. Taylor expanded in A around -inf 53.9

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

   if 8.000000000000001e-30 < B

   1. Initial program 53.5

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

   2. Simplified 51.2

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

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

   3. Applied egg-rr 46.9

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

   4. Simplified 46.9

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

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

   5. Applied egg-rr 46.9

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

   6. Simplified 46.9

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

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

   7. Taylor expanded in A around 0 33.5

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

   8. Simplified 33.5

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

      [Start] 33.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      mul-1-neg [=>] 33.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \]
      distribute-neg-frac [=>] 33.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2}}{B}} \]

3. Recombined 7 regimes into one program.

4. Final simplification 42.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -7 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-136}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{C}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-203}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.2
Cost	47052

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F \cdot \left(C + \left(A - t_1\right)\right)}\\ t_4 := -\sqrt{2 \cdot t_0}\\ \mathbf{if}\;B \leq -1.18 \cdot 10^{+24}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + \left(C - t_1\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot t_2}}{t_2}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-203}:\\ \;\;\;\;\frac{t_4 \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{A}\right)\right)}\right)}^{2}}{t_2}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)} \cdot \frac{t_4}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	41.7
Cost	34836

\[\begin{array}{l} t_0 := \frac{B \cdot B}{A - C}\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{-18}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \mathsf{fma}\left(-0.5, t_0, C \cdot 2\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 10^{-204}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot t_0\right)\right)} \cdot \frac{-\sqrt{2 \cdot t_2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	41.1
Cost	20688

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-204}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	42.6
Cost	20228

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -3.7 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \left(C - B\right)\right)}\\ \end{array} \]

Alternative 5
Error	43.9
Cost	15632

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-179}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-204}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \left(C - B\right)\right)}\\ \end{array} \]

Alternative 6
Error	49.2
Cost	13964

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \left(C - B\right)\right)}\\ \end{array} \]

Alternative 7
Error	44.7
Cost	13964

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.42 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \left(C - B\right)\right)}\\ \end{array} \]

Alternative 8
Error	51.3
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ t_2 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;C \leq -1.3 \cdot 10^{+197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -1.85 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.32 \cdot 10^{-304}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{+82}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 9
Error	51.4
Cost	8456

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{+197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -2.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 10
Error	54.4
Cost	8072

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;C \leq -7.6 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 9.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B + C\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 11
Error	55.5
Cost	6921

\[\begin{array}{l} \mathbf{if}\;A \leq -1.8 \cdot 10^{-236} \lor \neg \left(A \leq 10^{+47}\right):\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 12
Error	56.7
Cost	6656

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

Alternative 13
Error	63.3
Cost	6592

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

Reproduce

herbie shell --seed 2023045 
(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))))