ABCF->ab-angle b

Average Error: 52.3 → 39.8

Time: 43.2s

Precision: binary64

Cost: 27916

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -8.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 62000000000:\\ \;\;\;\;\frac{-{\left({\left(2 \cdot \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), -8 \cdot \left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, \frac{C}{\frac{B}{A}}, B\right)}\\ \end{array} \]

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 (- (sqrt (* 2.0 (* F (+ C (- A (hypot B (- A C)))))))))
        (t_1 (fma A (* C -4.0) (* B B))))
   (if (<= B -8.8e-85)
     (/ t_0 (- B))
     (if (<= B 1.2e-196)
       (/
        (- (sqrt (* t_1 (* (+ A (+ A (* -0.5 (/ (* B B) C)))) (* 2.0 F)))))
        t_1)
       (if (<= B 62000000000.0)
         (/
          (-
           (pow
            (pow
             (* 2.0 (fma 2.0 (* C (* F (* B B))) (* -8.0 (* (* F C) (* C A)))))
             0.25)
            2.0))
          (fma B B (* A (* C -4.0))))
         (/ t_0 (fma -2.0 (/ C (/ B A)) 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 = -sqrt((2.0 * (F * (C + (A - hypot(B, (A - C)))))));
	double t_1 = fma(A, (C * -4.0), (B * B));
	double tmp;
	if (B <= -8.8e-85) {
		tmp = t_0 / -B;
	} else if (B <= 1.2e-196) {
		tmp = -sqrt((t_1 * ((A + (A + (-0.5 * ((B * B) / C)))) * (2.0 * F)))) / t_1;
	} else if (B <= 62000000000.0) {
		tmp = -pow(pow((2.0 * fma(2.0, (C * (F * (B * B))), (-8.0 * ((F * C) * (C * A))))), 0.25), 2.0) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = t_0 / fma(-2.0, (C / (B / A)), B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(C + Float64(A - hypot(B, Float64(A - C))))))))
	t_1 = fma(A, Float64(C * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= -8.8e-85)
		tmp = Float64(t_0 / Float64(-B));
	elseif (B <= 1.2e-196)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + Float64(A + Float64(-0.5 * Float64(Float64(B * B) / C)))) * Float64(2.0 * F))))) / t_1);
	elseif (B <= 62000000000.0)
		tmp = Float64(Float64(-((Float64(2.0 * fma(2.0, Float64(C * Float64(F * Float64(B * B))), Float64(-8.0 * Float64(Float64(F * C) * Float64(C * A))))) ^ 0.25) ^ 2.0)) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(t_0 / fma(-2.0, Float64(C / Float64(B / A)), 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[N[(2.0 * N[(F * N[(C + N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -8.8e-85], N[(t$95$0 / (-B)), $MachinePrecision], If[LessEqual[B, 1.2e-196], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + N[(A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 62000000000.0], N[((-N[Power[N[Power[N[(2.0 * N[(2.0 * N[(C * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[(N[(F * C), $MachinePrecision] * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(-2.0 * N[(C / N[(B / A), $MachinePrecision]), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -8.8e-85

   1. Initial program 52.9

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

   2. Simplified 50.6

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

      [Start] 52.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.9

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

   4. Simplified 46.7

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

      [Start] 46.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 46.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 46.9

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

   6. Simplified 45.8

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

      [Start] 46.9	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 46.9	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 45.9	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 45.9	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 45.9	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 45.9	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-*l* [=>] 45.9	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      +-commutative [=>] 45.9	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate--l+ [=>] 45.8	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around -inf 33.7

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

   8. Simplified 33.7

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

      [Start] 33.7	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-1 \cdot B} \]
      mul-1-neg [=>] 33.7	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{-B}} \]

   if -8.8e-85 < B < 1.2000000000000001e-196

   1. Initial program 52.5

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

   2. Simplified 46.9

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

      [Start] 52.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 51.1

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

   4. Simplified 51.3

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

      [Start] 51.1	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 51.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\color{blue}{{B}^{2} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 51.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\color{blue}{B \cdot B} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 51.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{B \cdot B + \left(\color{blue}{A \cdot A} - {\left(-1 \cdot A\right)}^{2}\right)}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 51.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{B \cdot B + \left(A \cdot A - {\color{blue}{\left(-A\right)}}^{2}\right)}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in B around 0 49.4

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

   6. Simplified 49.4

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

      [Start] 49.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 49.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 1.2000000000000001e-196 < B < 6.2e10

   1. Initial program 46.5

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

   2. Simplified 40.0

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

      [Start] 46.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 53.8

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

   4. Simplified 53.7

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

      [Start] 53.8	\[ \frac{-\sqrt{2 \cdot \left(2 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right) + -8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 53.8	\[ \frac{-\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(2, C \cdot \left(F \cdot {B}^{2}\right), -8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 53.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(2, \color{blue}{\left(C \cdot F\right) \cdot {B}^{2}}, -8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 53.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(2, \color{blue}{\left(F \cdot C\right)} \cdot {B}^{2}, -8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 53.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(2, \left(F \cdot C\right) \cdot \color{blue}{\left(B \cdot B\right)}, -8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 53.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(2, \left(F \cdot C\right) \cdot \left(B \cdot B\right), -8 \cdot \left(A \cdot \color{blue}{\left(F \cdot {C}^{2}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 53.7	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(2, \left(F \cdot C\right) \cdot \left(B \cdot B\right), -8 \cdot \left(A \cdot \left(F \cdot \color{blue}{\left(C \cdot C\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 48.7

      \[\leadsto \frac{-\color{blue}{{\left({\left(2 \cdot \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), -8 \cdot \left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 6.2e10 < B

   1. Initial program 55.5

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

   2. Simplified 53.8

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

      [Start] 55.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 48.7

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

   4. Simplified 48.6

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

      [Start] 48.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 48.6	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 48.7

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

   6. Simplified 47.4

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

      [Start] 48.7	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 48.7	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 47.5	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 47.5	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 47.5	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 47.5	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-*l* [=>] 47.5	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      +-commutative [=>] 47.5	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate--l+ [=>] 47.4	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around 0 32.2

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

   8. Simplified 31.3

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

      [Start] 32.2	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-2 \cdot \frac{A \cdot C}{B} + B} \]
      fma-def [=>] 32.2	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-2, \frac{A \cdot C}{B}, B\right)}} \]
      *-commutative [=>] 32.2	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-2, \frac{\color{blue}{C \cdot A}}{B}, B\right)} \]
      associate-/l* [=>] 31.3	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-2, \color{blue}{\frac{C}{\frac{B}{A}}}, B\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 39.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 62000000000:\\ \;\;\;\;\frac{-{\left({\left(2 \cdot \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), -8 \cdot \left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right)\right)\right)}^{0.25}\right)}^{2}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-2, \frac{C}{\frac{B}{A}}, B\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	39.8
Cost	21516

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 85000000000:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), -8 \cdot \left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, \frac{C}{\frac{B}{A}}, B\right)}\\ \end{array} \]

Alternative 2
Error	40.2
Cost	21384

\[\begin{array}{l} t_0 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \frac{F}{\frac{1}{t_0}}}}{-B}\\ \mathbf{elif}\;B \leq 62000000000:\\ \;\;\;\;\sqrt{2 \cdot \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), -8 \cdot \left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{\mathsf{fma}\left(-2, \frac{C}{\frac{B}{A}}, B\right)}\\ \end{array} \]

Alternative 3
Error	40.3
Cost	20808

\[\begin{array}{l} t_0 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \frac{F}{\frac{1}{t_0}}}}{-B}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot \left(C + C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{\mathsf{fma}\left(-2, \frac{C}{\frac{B}{A}}, B\right)}\\ \end{array} \]

Alternative 4
Error	40.3
Cost	14344

\[\begin{array}{l} t_0 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \frac{F}{\frac{1}{t_0}}}}{-B}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot \left(C + C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{B}\\ \end{array} \]

Alternative 5
Error	41.4
Cost	14216

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-85}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 1.36 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 6
Error	42.7
Cost	14024

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 7
Error	41.7
Cost	14024

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -5.4 \cdot 10^{-85}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 8
Error	49.2
Cost	13700

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 9
Error	49.1
Cost	13700

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 10
Error	50.0
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(C + A\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 11
Error	50.1
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(C + A\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 12
Error	54.2
Cost	8452

\[\begin{array}{l} \mathbf{if}\;B \leq -5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(C + A\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot C\right) \cdot \left(B \cdot B + C \cdot \left(A \cdot -4 + A \cdot -8\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 13
Error	51.9
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 3.1 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	54.4
Cost	7940

\[\begin{array}{l} \mathbf{if}\;B \leq -7.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B + \left(C + A\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A \cdot \left(-4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 15
Error	55.1
Cost	7236

\[\begin{array}{l} \mathbf{if}\;C \leq -2.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + C\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 16
Error	55.2
Cost	6976

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

Reproduce

herbie shell --seed 2023088 
(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))))