ABCF->ab-angle b

Average Accuracy: 17.8% → 39.0%

Time: 59.0s

Precision: binary64

Cost: 34384

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := F \cdot \left(C - \left(t_0 - A\right)\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;t_3 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot t_1\right)}}{t_4}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_4}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_2} \cdot \left(-\sqrt{t_1}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (* F (- C (- t_0 A))))
        (t_2 (fma B B (* C (* A -4.0))))
        (t_3 (sqrt (* F (+ A (- C t_0)))))
        (t_4 (fma B B (* A (* C -4.0)))))
   (if (<= B -2.6e+20)
     (* t_3 (/ (sqrt 2.0) B))
     (if (<= B -9.8e-133)
       (/ (- (sqrt (* 2.0 (* t_4 t_1)))) t_4)
       (if (<= B 8.5e-160)
         (/ (- (sqrt (* (* 2.0 (* F (* A C))) (* A -8.0)))) t_4)
         (if (<= B 1.05e+50)
           (/ (* (sqrt (* 2.0 t_2)) (- (sqrt t_1))) t_2)
           (* 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 = F * (C - (t_0 - A));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double t_3 = sqrt((F * (A + (C - t_0))));
	double t_4 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -2.6e+20) {
		tmp = t_3 * (sqrt(2.0) / B);
	} else if (B <= -9.8e-133) {
		tmp = -sqrt((2.0 * (t_4 * t_1))) / t_4;
	} else if (B <= 8.5e-160) {
		tmp = -sqrt(((2.0 * (F * (A * C))) * (A * -8.0))) / t_4;
	} else if (B <= 1.05e+50) {
		tmp = (sqrt((2.0 * t_2)) * -sqrt(t_1)) / t_2;
	} 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 = Float64(F * Float64(C - Float64(t_0 - A)))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_3 = sqrt(Float64(F * Float64(A + Float64(C - t_0))))
	t_4 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -2.6e+20)
		tmp = Float64(t_3 * Float64(sqrt(2.0) / B));
	elseif (B <= -9.8e-133)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * t_1)))) / t_4);
	elseif (B <= 8.5e-160)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * Float64(A * C))) * Float64(A * -8.0)))) / t_4);
	elseif (B <= 1.05e+50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_2)) * Float64(-sqrt(t_1))) / t_2);
	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[(F * N[(C - N[(t$95$0 - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $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]}, If[LessEqual[B, -2.6e+20], N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -9.8e-133], N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 8.5e-160], N[((-N[Sqrt[N[(N[(2.0 * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 1.05e+50], N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$3 * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -2.6e20

   1. Initial program 12.2%

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

   2. Simplified 14.6%

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

      [Start] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 22.5%

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

   4. Simplified 22.7%

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

      [Start] 22.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 22.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [<=] 22.5	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(A + C\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r- [<=] 22.7	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 22.6%

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

   6. Simplified 22.7%

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

      [Start] 22.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      associate--r- [=>] 22.7	\[ \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 22.7	\[ \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      distribute-frac-neg [<=] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      *-commutative [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*l* [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(-4 \cdot C\right)}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 22.7	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(-4 \cdot C\right)}\right)} \]

   7. Taylor expanded in B around -inf 49.4%

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

   if -2.6e20 < B < -9.79999999999999992e-133

   1. Initial program 29.3%

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

   2. Simplified 37.9%

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

      [Start] 29.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -9.79999999999999992e-133 < B < 8.49999999999999959e-160

   1. Initial program 16.4%

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

   2. Simplified 25.7%

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

      [Start] 16.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 19.0%

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

   4. Simplified 19.1%

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

      [Start] 19.0	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 19.1	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(-4 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 19.1	\[ \frac{-\sqrt{2 \cdot \left(\left(-4 \cdot A\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 22.8%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(-8 \cdot A\right)} + 0\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   6. Simplified 22.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      Proof

      [Start] 22.8	\[ \frac{-\left(\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(-8 \cdot A\right)} + 0\right)}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-rgt-identity [=>] 22.8	\[ \frac{-\color{blue}{\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(-8 \cdot A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 19.1	\[ \frac{-\sqrt{\color{blue}{\left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)} \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      distribute-rgt-out [<=] 19.1	\[ \frac{-\sqrt{\color{blue}{\left(A \cdot \left(F \cdot C\right) + A \cdot \left(F \cdot C\right)\right)} \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      count-2 [=>] 19.1	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)} \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.1	\[ \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(\left(F \cdot C\right) \cdot A\right)}\right) \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(F \cdot \left(C \cdot A\right)\right)}\right) \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [<=] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\left(A \cdot C\right)}\right)\right) \cdot \left(-8 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \color{blue}{\left(A \cdot -8\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 8.49999999999999959e-160 < B < 1.05e50

   1. Initial program 29.7%

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

   2. Simplified 38.9%

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

      [Start] 29.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 34.8%

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

   4. Simplified 35.7%

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

      [Start] 34.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 34.8	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate--l+ [=>] 35.7	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.05e50 < B

   1. Initial program 9.5%

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

   2. Simplified 12.1%

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

      [Start] 9.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 20.5%

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

   4. Simplified 20.6%

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

      [Start] 20.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 20.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [<=] 20.5	\[ \frac{-\sqrt{F \cdot \left(\color{blue}{\left(A + C\right)} - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r- [<=] 20.6	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Applied egg-rr 20.5%

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

   6. Simplified 20.5%

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

      [Start] 20.5	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      associate--r- [=>] 20.5	\[ \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 20.5	\[ \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      distribute-lft-neg-out [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right)} \]
      associate-*r/ [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot 1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\right) \]
      *-rgt-identity [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\right) \]
      distribute-frac-neg [<=] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      *-commutative [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*l* [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(-4 \cdot C\right)}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)} \]
      associate-*l* [=>] 20.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(-4 \cdot C\right)}\right)} \]

   7. Taylor expanded in B around inf 51.5%

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

   8. Simplified 51.5%

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

      [Start] 51.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      associate-*r/ [=>] 51.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \]
      mul-1-neg [=>] 51.5	\[ \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\color{blue}{-\sqrt{2}}}{B} \]

3. Recombined 5 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	39.0%
Cost	34384

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.22 \cdot 10^{+20}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C - \left(t_0 - A\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+69}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2 \cdot t_2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Accuracy	39.1%
Cost	34384

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+20}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(C - \left(t_0 - A\right)\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Accuracy	39.5%
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{F \cdot \left(A + \left(C - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C - \left(t_0 - A\right)\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -1.08 \cdot 10^{+21}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-132}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Accuracy	35.3%
Cost	27664

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_3 := -\sqrt{2}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t_2 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{t_3 \cdot \sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot t_1} \cdot \sqrt{F \cdot \left(A \cdot 2\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_3}{B}\\ \end{array} \]

Alternative 5
Accuracy	35.6%
Cost	27664

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{-12}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-67}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_1} \cdot \left(-\sqrt{F \cdot \left(C \cdot 2\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Accuracy	35.2%
Cost	20876

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t_1 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-53}:\\ \;\;\;\;\frac{t_2 \cdot \sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_2}{B}\\ \end{array} \]

Alternative 7
Accuracy	35.1%
Cost	20688

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t_0 \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Accuracy	25.7%
Cost	20432

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -4.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	34.2%
Cost	20432

\[\begin{array}{l} \mathbf{if}\;B \leq -4.4 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]

Alternative 10
Accuracy	22.5%
Cost	15764

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{\left(\left(A \cdot C\right) \cdot \left(F \cdot C\right)\right) \cdot -16}}{A \cdot \left(C \cdot 4\right)}\\ t_2 := B \cdot B + t_0\\ t_3 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_2}\\ \mathbf{if}\;C \leq -7.5 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -9.2 \cdot 10^{-193}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -4.1 \cdot 10^{-269}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 4.4 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(B + \frac{\left(C \cdot C\right) \cdot 0.5}{B}\right)\right)}\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-108}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot \left(A + A\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, t_0\right)}\\ \end{array} \]

Alternative 11
Accuracy	21.0%
Cost	14480

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(F \cdot C\right)\\ t_2 := \frac{\sqrt{A \cdot \left(-16 \cdot t_1\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{-12}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot t_1\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(B + \frac{\left(C \cdot C\right) \cdot 0.5}{B}\right)\right)}\\ \end{array} \]

Alternative 12
Accuracy	22.4%
Cost	14480

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(F \cdot C\right)\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot t_1\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot t_1\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(B + \frac{\left(C \cdot C\right) \cdot 0.5}{B}\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	20.3%
Cost	14348

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-212}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(B + \frac{\left(C \cdot C\right) \cdot 0.5}{B}\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	21.2%
Cost	14348

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.75 \cdot 10^{-198}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 3.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(B + \frac{\left(C \cdot C\right) \cdot 0.5}{B}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 15
Accuracy	19.9%
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 16
Accuracy	20.3%
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.35 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 17
Accuracy	13.3%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq 5.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(C - C\right) - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 18
Accuracy	18.7%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.4 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 19
Accuracy	13.8%
Cost	7680

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

Alternative 20
Accuracy	1.9%
Cost	7360

\[\frac{-1}{B} \cdot \frac{\sqrt{\left(A \cdot C\right) \cdot \left(-16 \cdot \left(F \cdot C\right)\right)}}{B} \]

Alternative 21
Accuracy	13.8%
Cost	7360

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

Alternative 22
Accuracy	1.9%
Cost	7296

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

Alternative 23
Accuracy	1.6%
Cost	7232

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

Reproduce

herbie shell --seed 2023122 
(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))))