ABCF->ab-angle b

Average Error: 52.4 → 40.0

Time: 58.0s

Precision: binary64

Cost: 47184

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := F \cdot \left(C + \left(A - t_0\right)\right)\\ t_3 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_5 := \sqrt{2 \cdot t_4}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+86}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -5.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{t_2}\right)}{t_4}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4 - \frac{B}{\frac{C}{B}}\right) \cdot \left(F \cdot t_1\right)}}{-t_1}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-128}:\\ \;\;\;\;\frac{-t_5}{t_4} \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{C}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(A \cdot \left(F \cdot 4\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 3700:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot t_2\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 B B (* C (* A -4.0))))
        (t_2 (* F (+ C (- A t_0))))
        (t_3 (sqrt (* F (+ A (- C t_0)))))
        (t_4 (fma B B (* A (* C -4.0))))
        (t_5 (sqrt (* 2.0 t_4))))
   (if (<= B -4.2e+86)
     (* t_3 (/ (sqrt 2.0) B))
     (if (<= B -5.6e-76)
       (/ (* t_5 (- (sqrt t_2))) t_4)
       (if (<= B 4.2e-259)
         (/ (sqrt (* (- (* A 4.0) (/ B (/ C B))) (* F t_1))) (- t_1))
         (if (<= B 1.05e-128)
           (*
            (/ (- t_5) t_4)
            (pow (exp (* 0.25 (- (log (* F -2.0)) (log (/ -1.0 C))))) 2.0))
           (if (<= B 1.2e-94)
             (/ (- (sqrt (* t_4 (* A (* F 4.0))))) t_4)
             (if (<= B 3700.0)
               (/ (- (sqrt (* 2.0 (* t_4 t_2)))) t_4)
               (* 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(B, B, (C * (A * -4.0)));
	double t_2 = F * (C + (A - t_0));
	double t_3 = sqrt((F * (A + (C - t_0))));
	double t_4 = fma(B, B, (A * (C * -4.0)));
	double t_5 = sqrt((2.0 * t_4));
	double tmp;
	if (B <= -4.2e+86) {
		tmp = t_3 * (sqrt(2.0) / B);
	} else if (B <= -5.6e-76) {
		tmp = (t_5 * -sqrt(t_2)) / t_4;
	} else if (B <= 4.2e-259) {
		tmp = sqrt((((A * 4.0) - (B / (C / B))) * (F * t_1))) / -t_1;
	} else if (B <= 1.05e-128) {
		tmp = (-t_5 / t_4) * pow(exp((0.25 * (log((F * -2.0)) - log((-1.0 / C))))), 2.0);
	} else if (B <= 1.2e-94) {
		tmp = -sqrt((t_4 * (A * (F * 4.0)))) / t_4;
	} else if (B <= 3700.0) {
		tmp = -sqrt((2.0 * (t_4 * t_2))) / t_4;
	} 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(B, B, Float64(C * Float64(A * -4.0)))
	t_2 = Float64(F * Float64(C + Float64(A - t_0)))
	t_3 = sqrt(Float64(F * Float64(A + Float64(C - t_0))))
	t_4 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_5 = sqrt(Float64(2.0 * t_4))
	tmp = 0.0
	if (B <= -4.2e+86)
		tmp = Float64(t_3 * Float64(sqrt(2.0) / B));
	elseif (B <= -5.6e-76)
		tmp = Float64(Float64(t_5 * Float64(-sqrt(t_2))) / t_4);
	elseif (B <= 4.2e-259)
		tmp = Float64(sqrt(Float64(Float64(Float64(A * 4.0) - Float64(B / Float64(C / B))) * Float64(F * t_1))) / Float64(-t_1));
	elseif (B <= 1.05e-128)
		tmp = Float64(Float64(Float64(-t_5) / t_4) * (exp(Float64(0.25 * Float64(log(Float64(F * -2.0)) - log(Float64(-1.0 / C))))) ^ 2.0));
	elseif (B <= 1.2e-94)
		tmp = Float64(Float64(-sqrt(Float64(t_4 * Float64(A * Float64(F * 4.0))))) / t_4);
	elseif (B <= 3700.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * t_2)))) / t_4);
	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[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(F * N[(A + N[(C - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -4.2e+86], N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -5.6e-76], N[(N[(t$95$5 * (-N[Sqrt[t$95$2], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 4.2e-259], N[(N[Sqrt[N[(N[(N[(A * 4.0), $MachinePrecision] - N[(B / N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B, 1.05e-128], N[(N[((-t$95$5) / t$95$4), $MachinePrecision] * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(F * -2.0), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.2e-94], N[((-N[Sqrt[N[(t$95$4 * N[(A * N[(F * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 3700.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], N[(t$95$3 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.1999999999999998e86

   1. Initial program 59.3

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

   2. Simplified 58.6

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

      [Start] 59.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 53.3

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

   4. Simplified 53.3

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

      [Start] 53.3	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 53.3	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+l- [<=] 53.3	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [<=] 53.3	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 53.3	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 63.1

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

   6. Simplified 53.3

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

      [Start] 63.1	\[ e^{\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} - 1 \]
      expm1-def [=>] 54.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 53.3	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-rgt-identity [<=] 53.3	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot 1} \]
      associate-*l/ [=>] 53.3	\[ \color{blue}{\frac{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      associate-*r/ [<=] 53.3	\[ \color{blue}{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 53.3	\[ \color{blue}{\left(\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [<=] 53.3	\[ \color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      distribute-lft-neg-out [=>] 53.3	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]

   7. Taylor expanded in B around -inf 30.8

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

   if -4.1999999999999998e86 < B < -5.6000000000000002e-76

   1. Initial program 42.0

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

   2. Simplified 36.4

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

      [Start] 42.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 35.8

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

   if -5.6000000000000002e-76 < B < 4.19999999999999995e-259

   1. Initial program 52.3

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

   2. Simplified 46.3

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

      [Start] 52.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 51.4

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

   4. Simplified 51.4

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

      [Start] 51.4	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 51.4	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(\color{blue}{B \cdot B} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 51.4	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + \color{blue}{A \cdot A}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 51.4	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\color{blue}{\left(-A\right)}}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 51.4

      \[\leadsto \color{blue}{0 - \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Simplified 49.7

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

      [Start] 51.4	\[ 0 - \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sub0-neg [=>] 51.4	\[ \color{blue}{-\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      mul-1-neg [<=] 51.4	\[ \color{blue}{-1 \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      metadata-eval [<=] 51.4	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      times-frac [<=] 51.4	\[ \color{blue}{\frac{1 \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{-1 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-lft-identity [=>] 51.4	\[ \frac{\color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\left(-A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}}{-1 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 4.19999999999999995e-259 < B < 1.0500000000000001e-128

   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 47.2

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

      [Start] 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. Applied egg-rr 51.7

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

   4. Simplified 51.7

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

      [Start] 51.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.7	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+l- [<=] 51.7	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [<=] 51.7	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 51.7	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 62.9

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

   6. Simplified 51.6

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

      [Start] 62.9	\[ e^{\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} - 1 \]
      expm1-def [=>] 59.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 51.7	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-rgt-identity [<=] 51.7	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot 1} \]
      associate-*l/ [=>] 51.7	\[ \color{blue}{\frac{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      associate-*r/ [<=] 52.0	\[ \color{blue}{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 52.0	\[ \color{blue}{\left(\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [<=] 52.0	\[ \color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      distribute-lft-neg-out [=>] 52.0	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]

   7. Applied egg-rr 51.7

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

   8. Taylor expanded in C around -inf 53.3

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

   if 1.0500000000000001e-128 < B < 1.2e-94

   1. Initial program 50.6

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

   2. Simplified 45.2

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

      [Start] 50.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 49.5

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

   4. Simplified 49.5

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

      [Start] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(\color{blue}{B \cdot B} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + \color{blue}{A \cdot A}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left(B \cdot B + A \cdot A\right) - {\color{blue}{\left(-A\right)}}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Taylor expanded in A around inf 48.3

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

   6. Simplified 48.3

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

      [Start] 48.3	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 48.3	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \color{blue}{\left(\left(A \cdot F\right) \cdot 4\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 48.3	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \color{blue}{\left(A \cdot \left(F \cdot 4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 1.2e-94 < B < 3700

   1. Initial program 44.3

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

   2. Simplified 39.5

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

      [Start] 44.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 3700 < B

   1. Initial program 55.6

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

   2. Simplified 54.1

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

      [Start] 55.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.1

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

   4. Simplified 49.1

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

      [Start] 49.1	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+l- [<=] 49.1	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [<=] 49.1	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 62.8

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

   6. Simplified 49.1

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

      [Start] 62.8	\[ e^{\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} - 1 \]
      expm1-def [=>] 53.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)\right)} \]
      expm1-log1p [=>] 49.1	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-rgt-identity [<=] 49.1	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot 1} \]
      associate-*l/ [=>] 49.1	\[ \color{blue}{\frac{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      associate-*r/ [<=] 49.1	\[ \color{blue}{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 49.1	\[ \color{blue}{\left(\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [<=] 49.1	\[ \color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      distribute-lft-neg-out [=>] 49.1	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]

   7. Taylor expanded in B around inf 33.5

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

   8. Simplified 33.5

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

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

3. Recombined 7 regimes into one program.

4. Final simplification 40.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -5.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4 - \frac{B}{\frac{C}{B}}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{C}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(A \cdot \left(F \cdot 4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3700:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.0
Cost	59976

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := \sqrt{2 \cdot t_3}\\ \mathbf{if}\;B \leq -5.6 \cdot 10^{+149}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{t_4}{t_3} \cdot \left(-{\left({\left(e^{0.25}\right)}^{\left(\log \left(t_0 - \left(A + C\right)\right) - \log \left(\frac{-1}{F}\right)\right)}\right)}^{2}\right)\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4 - \frac{B}{\frac{C}{B}}\right) \cdot \left(F \cdot t_1\right)}}{-t_1}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{-t_4}{t_3} \cdot {\left(e^{0.25 \cdot \left(\log \left(F \cdot -2\right) - \log \left(\frac{-1}{C}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;B \leq 3700:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	40.1
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_2 := F \cdot \left(C + C\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := \frac{-\sqrt{t_3 \cdot \left(A \cdot \left(F \cdot 4\right)\right)}}{t_3}\\ t_5 := \sqrt{2 \cdot t_3}\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+89}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-70}:\\ \;\;\;\;\frac{t_5}{t_3} \cdot \left(-t_1\right)\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-154}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot t_2\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-211}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-102}:\\ \;\;\;\;\frac{-t_5}{t_3} \cdot \sqrt{t_2}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-92}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3700:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	39.1
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -8 \cdot 10^{+81}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_3} \cdot \left(-\sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\right)}{t_3}\\ \mathbf{elif}\;B \leq -2.95 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4 - \frac{B}{\frac{C}{B}}\right) \cdot \left(F \cdot t_1\right)}}{-t_1}\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(\left(t_0 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	39.0
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-154}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(A \cdot \left(F \cdot 4\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3700:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	40.5
Cost	20872

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A \cdot \left(F \cdot 4\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	41.3
Cost	20688

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ t_2 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -3.2 \cdot 10^{-70}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-275}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 4\right) \cdot \left(\left(F \cdot C\right) \cdot \left(\mathsf{hypot}\left(A, B\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	47.3
Cost	20432

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, C\right) - C\right) - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-275}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 4\right) \cdot \left(\left(F \cdot C\right) \cdot \left(\mathsf{hypot}\left(A, B\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 8
Error	41.9
Cost	20432

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -7.1 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -2.85 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.1 \cdot 10^{-274}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 4\right) \cdot \left(\left(F \cdot C\right) \cdot \left(\mathsf{hypot}\left(A, B\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 9
Error	47.8
Cost	15380

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, C\right) - C\right) - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ t_2 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-275}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 4\right) \cdot \left(\left(F \cdot C\right) \cdot \left(\mathsf{hypot}\left(A, B\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.68 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 10
Error	48.8
Cost	14796

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{-66}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-275}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 4\right) \cdot \left(\left(F \cdot C\right) \cdot \left(\mathsf{hypot}\left(A, B\right) - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \left(A - B\right)\right)}\right)\\ \end{array} \]

Alternative 11
Error	48.7
Cost	14096

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{-65}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-273}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \left(A - B\right)\right)}\right)\\ \end{array} \]

Alternative 12
Error	48.8
Cost	13968

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-63}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-274}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 13
Error	48.7
Cost	13904

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-63}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.22 \cdot 10^{-268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-275}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 14
Error	53.0
Cost	8848

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ t_2 := -\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -6.8 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -2.9 \cdot 10^{-279}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 1.6 \cdot 10^{-228}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 5.7 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	53.6
Cost	8068

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot 8\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	52.8
Cost	8068

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{-143}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	54.8
Cost	7940

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot 8\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A \cdot \left(-8 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	58.5
Cost	7808

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

Alternative 19
Error	57.2
Cost	7808

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

Alternative 20
Error	62.6
Cost	7680

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

Alternative 21
Error	62.7
Cost	7616

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

Alternative 22
Error	63.1
Cost	7424

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

Alternative 23
Error	63.1
Cost	7296

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

Reproduce

herbie shell --seed 2023011 
(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))))