ABCF->ab-angle b

Average Accuracy: 17.5% → 39.1%

Time: 54.2s

Precision: binary64

Cost: 28176

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_3 := \frac{B}{\sqrt{2}}\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+57}:\\ \;\;\;\;t_2 \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \left(2 \cdot \frac{C}{B}\right) - t_3}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(t_0 - C\right) - A\right)\right)} \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{+40}:\\ \;\;\;\;t_2 \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2}}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-1}{t_3}\\ \end{array} \]

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 (* A (* C -4.0))))
        (t_2 (sqrt (* F (+ C (- A t_0)))))
        (t_3 (/ B (sqrt 2.0))))
   (if (<= B -2.1e+57)
     (* t_2 (/ -1.0 (- (* (/ A (sqrt 2.0)) (* 2.0 (/ C B))) t_3)))
     (if (<= B 2.1e-145)
       (* (sqrt (* -2.0 (* (* F t_1) (- (- t_0 C) A)))) (/ 1.0 (- t_1)))
       (if (<= B 8.4e+40)
         (* t_2 (/ -1.0 (sqrt (/ (fma A (* C -4.0) (* B B)) 2.0))))
         (if (<= B 2.6e+49)
           (/
            (-
             (sqrt
              (* 2.0 (* t_1 (* F (fma -0.5 (/ (* B B) (- A C)) (* C 2.0)))))))
            t_1)
           (* t_2 (/ -1.0 t_3))))))))

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, (A * (C * -4.0)));
	double t_2 = sqrt((F * (C + (A - t_0))));
	double t_3 = B / sqrt(2.0);
	double tmp;
	if (B <= -2.1e+57) {
		tmp = t_2 * (-1.0 / (((A / sqrt(2.0)) * (2.0 * (C / B))) - t_3));
	} else if (B <= 2.1e-145) {
		tmp = sqrt((-2.0 * ((F * t_1) * ((t_0 - C) - A)))) * (1.0 / -t_1);
	} else if (B <= 8.4e+40) {
		tmp = t_2 * (-1.0 / sqrt((fma(A, (C * -4.0), (B * B)) / 2.0)));
	} else if (B <= 2.6e+49) {
		tmp = -sqrt((2.0 * (t_1 * (F * fma(-0.5, ((B * B) / (A - C)), (C * 2.0)))))) / t_1;
	} else {
		tmp = t_2 * (-1.0 / t_3);
	}
	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(A * Float64(C * -4.0)))
	t_2 = sqrt(Float64(F * Float64(C + Float64(A - t_0))))
	t_3 = Float64(B / sqrt(2.0))
	tmp = 0.0
	if (B <= -2.1e+57)
		tmp = Float64(t_2 * Float64(-1.0 / Float64(Float64(Float64(A / sqrt(2.0)) * Float64(2.0 * Float64(C / B))) - t_3)));
	elseif (B <= 2.1e-145)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(Float64(F * t_1) * Float64(Float64(t_0 - C) - A)))) * Float64(1.0 / Float64(-t_1)));
	elseif (B <= 8.4e+40)
		tmp = Float64(t_2 * Float64(-1.0 / sqrt(Float64(fma(A, Float64(C * -4.0), Float64(B * B)) / 2.0))));
	elseif (B <= 2.6e+49)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_1 * Float64(F * fma(-0.5, Float64(Float64(B * B) / Float64(A - C)), Float64(C * 2.0))))))) / t_1);
	else
		tmp = Float64(t_2 * Float64(-1.0 / t_3));
	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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(B / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.1e+57], N[(t$95$2 * N[(-1.0 / N[(N[(N[(A / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.1e-145], N[(N[Sqrt[N[(-2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(N[(t$95$0 - C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$1)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 8.4e+40], N[(t$95$2 * N[(-1.0 / N[Sqrt[N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.6e+49], N[((-N[Sqrt[N[(2.0 * N[(t$95$1 * N[(F * N[(-0.5 * N[(N[(B * B), $MachinePrecision] / N[(A - C), $MachinePrecision]), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(t$95$2 * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -2.09999999999999991e57

   1. Initial program 8.9%

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

   2. Simplified 10.7%

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

      [Start] 8.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 19.0%

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

      [Start] 10.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 10.7	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 19.0	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 19.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 19.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 18.9%

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

      [Start] 19.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.0	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+r- [=>] 18.9	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 18.9	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 19.0%

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

      [Start] 18.9	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      div-inv [=>] 18.9	\[ \color{blue}{\left(-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 18.9	\[ \color{blue}{\left(\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 18.9	\[ \color{blue}{\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      +-commutative [=>] 18.9	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      associate--l+ [=>] 19.0	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      *-commutative [=>] 19.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}\right) \]

   6. Applied egg-rr 19.0%

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

      [Start] 19.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      un-div-inv [=>] 19.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      neg-mul-1 [=>] 19.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\color{blue}{-1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 19.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]

   7. Taylor expanded in B around -inf 47.0%

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

   8. Simplified 49.4%

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

      [Start] 47.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{2 \cdot \frac{A \cdot C}{\sqrt{2} \cdot B} + -1 \cdot \frac{B}{\sqrt{2}}} \]
      mul-1-neg [=>] 47.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{2 \cdot \frac{A \cdot C}{\sqrt{2} \cdot B} + \color{blue}{\left(-\frac{B}{\sqrt{2}}\right)}} \]
      unsub-neg [=>] 47.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{2 \cdot \frac{A \cdot C}{\sqrt{2} \cdot B} - \frac{B}{\sqrt{2}}}} \]
      *-commutative [=>] 47.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\frac{A \cdot C}{\sqrt{2} \cdot B} \cdot 2} - \frac{B}{\sqrt{2}}} \]
      times-frac [=>] 49.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\left(\frac{A}{\sqrt{2}} \cdot \frac{C}{B}\right)} \cdot 2 - \frac{B}{\sqrt{2}}} \]
      associate-*l* [=>] 49.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\frac{A}{\sqrt{2}} \cdot \left(\frac{C}{B} \cdot 2\right)} - \frac{B}{\sqrt{2}}} \]
      *-commutative [=>] 49.4	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \color{blue}{\left(2 \cdot \frac{C}{B}\right)} - \frac{B}{\sqrt{2}}} \]

   if -2.09999999999999991e57 < B < 2.09999999999999991e-145

   1. Initial program 21.2%

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

   2. Simplified 30.8%

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

      [Start] 21.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 30.1%

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

      [Start] 30.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      frac-2neg [=>] 30.8	\[ \color{blue}{\frac{-\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}\right)}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      remove-double-neg [=>] 30.8	\[ \frac{\color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      div-inv [=>] 30.5	\[ \color{blue}{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      associate-*r* [=>] 30.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 30.3	\[ \sqrt{2 \cdot \color{blue}{\left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)}} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 30.1	\[ \sqrt{2 \cdot \left(\color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)} \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 30.1	\[ \sqrt{2 \cdot \left(\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 2.09999999999999991e-145 < B < 8.4000000000000004e40

   1. Initial program 30.2%

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

   2. Simplified 38.3%

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

      [Start] 30.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 35.8%

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

      [Start] 38.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 38.3	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 35.8	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 35.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 35.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 35.0%

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

      [Start] 35.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 35.8	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+r- [=>] 35.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 35.0	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 35.8%

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

      [Start] 35.0	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      div-inv [=>] 35.0	\[ \color{blue}{\left(-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 35.0	\[ \color{blue}{\left(\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 35.0	\[ \color{blue}{\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      +-commutative [=>] 35.0	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      associate--l+ [=>] 35.8	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      *-commutative [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}\right) \]

   6. Applied egg-rr 35.8%

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

      [Start] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      un-div-inv [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      neg-mul-1 [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\color{blue}{-1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]

   7. Applied egg-rr 28.6%

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

      [Start] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      add-sqr-sqrt [=>] 35.7	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}}} \]
      sqrt-unprod [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}}} \]
      frac-times [=>] 28.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}}} \]
      pow2 [=>] 28.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]
      *-commutative [=>] 28.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot -4\right) \cdot A}\right)\right)}^{2}}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]
      associate-*l* [=>] 28.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)\right)}^{2}}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]
      add-sqr-sqrt [<=] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{2}}{\color{blue}{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]
      *-commutative [=>] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{2}}{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot 2}}}} \]
      *-commutative [=>] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot -4\right) \cdot A}\right) \cdot 2}}} \]
      associate-*l* [=>] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right) \cdot 2}}} \]

   8. Simplified 36.0%

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

      [Start] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot 2}}} \]
      unpow2 [=>] 28.6	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot 2}}} \]
      associate-/l* [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot 2}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}}}}} \]
      *-commutative [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}{\frac{\color{blue}{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}}}} \]
      associate-/l* [=>] 35.8	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}{\color{blue}{\frac{2}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}}}}}} \]
      *-inverses [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}{\frac{2}{\color{blue}{1}}}}} \]
      fma-udef [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{B \cdot B + C \cdot \left(-4 \cdot A\right)}}{\frac{2}{1}}}} \]
      +-commutative [<=] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{C \cdot \left(-4 \cdot A\right) + B \cdot B}}{\frac{2}{1}}}} \]
      associate-*r* [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{\left(C \cdot -4\right) \cdot A} + B \cdot B}{\frac{2}{1}}}} \]
      *-commutative [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{A \cdot \left(C \cdot -4\right)} + B \cdot B}{\frac{2}{1}}}} \]
      fma-def [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\frac{2}{1}}}} \]
      metadata-eval [=>] 36.0	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\color{blue}{2}}}} \]

   if 8.4000000000000004e40 < B < 2.59999999999999989e49

   1. Initial program 35.2%

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

   2. Simplified 45.6%

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

      [Start] 35.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 17.1%

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

   4. Simplified 17.1%

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

      [Start] 17.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A - C} + 2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 17.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A - C}, 2 \cdot C\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 17.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A - C}, 2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 2.59999999999999989e49 < B

   1. Initial program 9.1%

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

   2. Simplified 11.3%

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

      [Start] 9.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 20.5%

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

      [Start] 11.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 11.3	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 20.5	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 20.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 20.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 20.4%

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

      [Start] 20.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 20.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+r- [=>] 20.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 20.4	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 20.5%

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

      [Start] 20.4	\[ \frac{-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      div-inv [=>] 20.4	\[ \color{blue}{\left(-\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 20.4	\[ \color{blue}{\left(\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 20.4	\[ \color{blue}{\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      +-commutative [=>] 20.4	\[ \sqrt{F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      associate--l+ [=>] 20.5	\[ \sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      *-commutative [=>] 20.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}\right) \]

   6. Applied egg-rr 20.5%

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

      [Start] 20.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      un-div-inv [=>] 20.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      neg-mul-1 [=>] 20.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\color{blue}{-1 \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 20.5	\[ \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}} \]

   7. Taylor expanded in B around inf 50.2%

      \[\leadsto \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\color{blue}{\frac{B}{\sqrt{2}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.1 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \left(2 \cdot \frac{C}{B}\right) - \frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-145}:\\ \;\;\;\;\sqrt{-2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2}}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	39.9%
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\\ t_2 := \sqrt{t_1}\\ t_3 := \frac{B}{\sqrt{2}}\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{+54}:\\ \;\;\;\;t_2 \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \left(2 \cdot \frac{C}{B}\right) - t_3}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-1}{t_3}\\ \end{array} \]

Alternative 2
Accuracy	39.8%
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(C + \left(A - t_0\right)\right)}\\ t_3 := \frac{B}{\sqrt{2}}\\ \mathbf{if}\;B \leq -1.75 \cdot 10^{+55}:\\ \;\;\;\;t_2 \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \left(2 \cdot \frac{C}{B}\right) - t_3}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(F \cdot -2\right) \cdot \left(\left(t_0 - C\right) - A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-1}{t_3}\\ \end{array} \]

Alternative 3
Accuracy	38.0%
Cost	27536

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\frac{-B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+153}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Accuracy	38.0%
Cost	27536

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -1.76 \cdot 10^{-50}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\frac{A}{\sqrt{2}} \cdot \left(2 \cdot \frac{C}{B}\right) - \frac{B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 10^{-198}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+153}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\sqrt{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Accuracy	36.3%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(\left(F \cdot 2\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -2.06 \cdot 10^{-50}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\frac{-B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 155000000 \lor \neg \left(B \leq 1.8 \cdot 10^{+26}\right):\\ \;\;\;\;t_0 \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\\ \end{array} \]

Alternative 6
Accuracy	31.9%
Cost	20620

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 7
Accuracy	34.7%
Cost	20620

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\frac{-B}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 8
Accuracy	31.9%
Cost	20556

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-198}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Accuracy	30.6%
Cost	20300

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-199}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	29.4%
Cost	14228

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + \left(C + A\right)\right)}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\left(C + A\right) - B\right)}\\ \end{array} \]

Alternative 11
Accuracy	19.9%
Cost	14096

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{if}\;B \leq 7.2 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\left(C + A\right) - B\right)}\\ \end{array} \]

Alternative 12
Accuracy	19.8%
Cost	13968

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{if}\;B \leq 4.1 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]

Alternative 13
Accuracy	19.9%
Cost	13904

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{if}\;B \leq 4.8 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ \end{array} \]

Alternative 14
Accuracy	12.8%
Cost	8848

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{if}\;B \leq 2.2 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Accuracy	18.8%
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.7 \cdot 10^{-111}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3 \cdot 10^{-53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \end{array} \]

Alternative 16
Accuracy	12.3%
Cost	8204

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -9.6 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 7.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{-2}{\frac{B}{\sqrt{F \cdot C}}}\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Accuracy	16.3%
Cost	8068

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Accuracy	15.3%
Cost	8068

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -2.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Accuracy	9.1%
Cost	7808

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

Alternative 20
Accuracy	9.1%
Cost	7680

\[-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)} \]

Alternative 21
Accuracy	3.0%
Cost	6848

\[\sqrt{F \cdot C} \cdot \frac{-2}{B} \]

Alternative 22
Accuracy	3.0%
Cost	6848

\[\frac{-2}{\frac{B}{\sqrt{F \cdot C}}} \]

Reproduce

herbie shell --seed 2023131 
(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))))