ABCF->ab-angle a

Average Error: 52.3 → 36.1

Time: 1.1min

Precision: binary64

Cost: 165640

Math

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

↓

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ t_2 := B \cdot \left(B \cdot F\right)\\ t_3 := B \cdot B + A \cdot A\\ t_4 := A + \sqrt{t_3}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ t_6 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_7 := \sqrt{\frac{1}{t_3}}\\ t_8 := 1 - A \cdot t_7\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\left(\sqrt{t_6} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2 \cdot \left(C + \left(A + t_5\right)\right)}}{t_6}\right)\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(2, t_2 \cdot t_4, 2 \cdot \left(C \cdot \mathsf{fma}\left(t_8, t_2, -4 \cdot \left(A \cdot \left(F \cdot t_4\right)\right)\right) + \left(C \cdot C\right) \cdot \mathsf{fma}\left(-4, F \cdot \left(A \cdot t_8\right), 0.5 \cdot \left(\left(1 - {\left(\left(-A\right) \cdot t_7\right)}^{2}\right) \cdot \left(t_2 \cdot t_7\right)\right)\right)\right)\right)}}{t_6}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_5\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\right)}{B \cdot B + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

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 (* -4.0 (* A C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* -2.0 (* F (- (* (* 4.0 A) C) (pow B 2.0))))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          (+ (pow B 2.0) (* C (* A -4.0)))))
        (t_2 (* B (* B F)))
        (t_3 (+ (* B B) (* A A)))
        (t_4 (+ A (sqrt t_3)))
        (t_5 (hypot B (- A C)))
        (t_6 (fma -4.0 (* A C) (* B B)))
        (t_7 (sqrt (/ 1.0 t_3)))
        (t_8 (- 1.0 (* A t_7))))
   (if (<= t_1 -2e-112)
     (* (* (sqrt t_6) (sqrt F)) (- (/ (sqrt (* 2.0 (+ C (+ A t_5)))) t_6)))
     (if (<= t_1 4e+73)
       (/
        (-
         (sqrt
          (fma
           2.0
           (* t_2 t_4)
           (*
            2.0
            (+
             (* C (fma t_8 t_2 (* -4.0 (* A (* F t_4)))))
             (*
              (* C C)
              (fma
               -4.0
               (* F (* A t_8))
               (* 0.5 (* (- 1.0 (pow (* (- A) t_7) 2.0)) (* t_2 t_7))))))))))
        t_6)
       (if (<= t_1 INFINITY)
         (/
          (* (sqrt (+ A (+ C t_5))) (- (sqrt (* 2.0 (* F (fma B B t_0))))))
          (+ (* B B) t_0))
         (* (/ (sqrt 2.0) B) (sqrt (* F (+ A (hypot B A))))))))))

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 = -4.0 * (A * C);
	double t_1 = -sqrt(((-2.0 * (F * (((4.0 * A) * C) - pow(B, 2.0)))) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (pow(B, 2.0) + (C * (A * -4.0)));
	double t_2 = B * (B * F);
	double t_3 = (B * B) + (A * A);
	double t_4 = A + sqrt(t_3);
	double t_5 = hypot(B, (A - C));
	double t_6 = fma(-4.0, (A * C), (B * B));
	double t_7 = sqrt((1.0 / t_3));
	double t_8 = 1.0 - (A * t_7);
	double tmp;
	if (t_1 <= -2e-112) {
		tmp = (sqrt(t_6) * sqrt(F)) * -(sqrt((2.0 * (C + (A + t_5)))) / t_6);
	} else if (t_1 <= 4e+73) {
		tmp = -sqrt(fma(2.0, (t_2 * t_4), (2.0 * ((C * fma(t_8, t_2, (-4.0 * (A * (F * t_4))))) + ((C * C) * fma(-4.0, (F * (A * t_8)), (0.5 * ((1.0 - pow((-A * t_7), 2.0)) * (t_2 * t_7))))))))) / t_6;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((A + (C + t_5))) * -sqrt((2.0 * (F * fma(B, B, t_0))))) / ((B * B) + t_0);
	} else {
		tmp = (sqrt(2.0) / B) * sqrt((F * (A + hypot(B, A))));
	}
	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(-4.0 * Float64(A * C))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(-2.0 * Float64(F * Float64(Float64(Float64(4.0 * A) * C) - (B ^ 2.0)))) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0))))
	t_2 = Float64(B * Float64(B * F))
	t_3 = Float64(Float64(B * B) + Float64(A * A))
	t_4 = Float64(A + sqrt(t_3))
	t_5 = hypot(B, Float64(A - C))
	t_6 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_7 = sqrt(Float64(1.0 / t_3))
	t_8 = Float64(1.0 - Float64(A * t_7))
	tmp = 0.0
	if (t_1 <= -2e-112)
		tmp = Float64(Float64(sqrt(t_6) * sqrt(F)) * Float64(-Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + t_5)))) / t_6)));
	elseif (t_1 <= 4e+73)
		tmp = Float64(Float64(-sqrt(fma(2.0, Float64(t_2 * t_4), Float64(2.0 * Float64(Float64(C * fma(t_8, t_2, Float64(-4.0 * Float64(A * Float64(F * t_4))))) + Float64(Float64(C * C) * fma(-4.0, Float64(F * Float64(A * t_8)), Float64(0.5 * Float64(Float64(1.0 - (Float64(Float64(-A) * t_7) ^ 2.0)) * Float64(t_2 * t_7)))))))))) / t_6);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + t_5))) * Float64(-sqrt(Float64(2.0 * Float64(F * fma(B, B, t_0)))))) / Float64(Float64(B * B) + t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A + hypot(B, A)))));
	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[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(N[(-2.0 * N[(F * N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * N[(B * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(B * B), $MachinePrecision] + N[(A * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(A + N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$6 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(1.0 - N[(A * t$95$7), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-112], N[(N[(N[Sqrt[t$95$6], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[N[(2.0 * N[(C + N[(A + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, 4e+73], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * t$95$4), $MachinePrecision] + N[(2.0 * N[(N[(C * N[(t$95$8 * t$95$2 + N[(-4.0 * N[(A * N[(F * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(C * C), $MachinePrecision] * N[(-4.0 * N[(F * N[(A * t$95$8), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(1.0 - N[Power[N[((-A) * t$95$7), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(A + N[(C + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.9999999999999999e-112

   1. Initial program 40.6

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

   2. Simplified 34.4

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

      [Start] 40.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 25.2

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

   4. Applied egg-rr 24.7

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

   5. Simplified 24.7

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

      [Start] 24.7	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 24.7	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 24.7	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 24.7	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 24.7	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \]

   6. Applied egg-rr 13.6

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

   if -1.9999999999999999e-112 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 3.99999999999999993e73

   1. Initial program 45.0

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

   2. Simplified 43.3

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

      [Start] 45.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 40.8

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

   4. Simplified 40.8

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

      [Start] 40.8

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

      fma-def [=>] 40.8

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

   if 3.99999999999999993e73 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 51.7

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

   2. Simplified 51.7

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

      [Start] 51.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 13.9

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

   4. Simplified 13.9

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

      [Start] 13.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 13.9	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 13.9	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 63.7

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

   4. Simplified 63.7

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

      [Start] 63.7	\[ \frac{-\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 63.7	\[ \frac{-\color{blue}{\sqrt{2} \cdot \left(B \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 63.7	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{\color{blue}{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 63.7	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 63.7	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 62.4

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

   6. Taylor expanded in C around 0 63.6

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]

   7. Simplified 53.7

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

      [Start] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      *-commutative [=>] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}} \]
      unpow2 [=>] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]
      unpow2 [=>] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \]
      hypot-def [=>] 53.7	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 36.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)\\ \mathbf{elif}\;\frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(2, \left(B \cdot \left(B \cdot F\right)\right) \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right), 2 \cdot \left(C \cdot \mathsf{fma}\left(1 - A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}, B \cdot \left(B \cdot F\right), -4 \cdot \left(A \cdot \left(F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right) + \left(C \cdot C\right) \cdot \mathsf{fma}\left(-4, F \cdot \left(A \cdot \left(1 - A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right), 0.5 \cdot \left(\left(1 - {\left(\left(-A\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)}^{2}\right) \cdot \left(\left(B \cdot \left(B \cdot F\right)\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.7
Cost	108616

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ t_2 := -\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\left(\sqrt{t_0} \cdot \sqrt{F}\right) \cdot t_2\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 2
Error	36.6
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+147}:\\ \;\;\;\;\left(-\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\right) \cdot \sqrt{F \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 3
Error	39.2
Cost	28128

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := A + \left(C + t_3\right)\\ t_5 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -4 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + t_3\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-198}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.34 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{t_4} \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{2 \cdot A} \cdot \left(-t_2\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_4}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{B \cdot \sqrt{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 4
Error	39.3
Cost	28128

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_4 := \frac{\sqrt{F \cdot t_3} \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_3}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -8.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + t_5\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-188}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A + \left(C + t_5\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{B \cdot \sqrt{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 5
Error	39.4
Cost	28128

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_3 := \sqrt{F \cdot t_2}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{2 \cdot \left(2 \cdot A\right)}\right)}{t_2}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + t_5\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-202}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot \left(2 \cdot C\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-188}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A + \left(C + t_5\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{B \cdot \sqrt{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 6
Error	36.6
Cost	27848

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\right)}{B \cdot B + t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 7
Error	37.9
Cost	27724

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{2 \cdot \left(C + \left(A + t_1\right)\right)}\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot t_2\right)}{B \cdot B + t_0}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-109}:\\ \;\;\;\;\left(-\frac{t_2}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right) \cdot \sqrt{F \cdot t_0}\\ \mathbf{elif}\;B \leq 1.56 \cdot 10^{+150}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A + \left(C + t_1\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{B \cdot \sqrt{F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 8
Error	40.5
Cost	21984

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B + t_1\\ t_3 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)}\\ t_4 := \frac{\sqrt{2 \cdot A} \cdot \left(-t_3\right)}{t_2}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -7 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-63}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + t_5\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq -1.85 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-203}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot C}\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-188}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.05 \cdot 10^{-10}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{-\sqrt{A + \left(C + t_5\right)} \cdot \sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9
Error	40.5
Cost	21984

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\\ t_4 := \mathsf{hypot}\left(B, A - C\right)\\ t_5 := \sqrt{A + \left(C + t_4\right)}\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq -6.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + t_4\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-198}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-182}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{2 \cdot A} \cdot \left(-t_3\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;\frac{-t_5 \cdot \sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10
Error	40.6
Cost	21788

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B + t_1\\ t_3 := \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)}\\ t_4 := \frac{\sqrt{2 \cdot A} \cdot \left(-t_3\right)}{t_2}\\ t_5 := \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot t_5\\ \mathbf{elif}\;B \leq -1.45 \cdot 10^{-64}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq -2.55 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{2 \cdot C}\right)}{t_2}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-188}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-10}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{t_5 \cdot \left(B \cdot \sqrt{2}\right)}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11
Error	40.7
Cost	21392

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot t_2\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{2 \cdot A} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{t_2 \cdot \left(B \cdot \sqrt{2}\right)}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12
Error	41.0
Cost	21140

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -1.62 \cdot 10^{+149}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot t_1\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{B \cdot \left(\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-105}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;B \leq 1.56 \cdot 10^{+150}:\\ \;\;\;\;\frac{t_1 \cdot \left(B \cdot \sqrt{2}\right)}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13
Error	41.6
Cost	21008

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -9.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot t_0\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-105}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{+148}:\\ \;\;\;\;\frac{t_0 \cdot \left(B \cdot \sqrt{2}\right)}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 14
Error	41.3
Cost	19972

\[\begin{array}{l} t_0 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{if}\;B \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-108}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 15
Error	45.2
Cost	15440

\[\begin{array}{l} t_0 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ t_1 := \left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 16
Error	47.8
Cost	15056

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -4 \cdot 10^{-101}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C + 2 \cdot A\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.65 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1050000000000:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 7 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	47.9
Cost	15056

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;C \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C + 2 \cdot A\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 4.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 800000000000:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;C \leq 7.4 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	47.4
Cost	14348

\[\begin{array}{l} \mathbf{if}\;C \leq -7 \cdot 10^{-152}:\\ \;\;\;\;\left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{elif}\;C \leq 1.65 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq 58000000:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(\left(A - C\right) - \left(A + C\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 19
Error	49.4
Cost	13772

\[\begin{array}{l} t_0 := \left(\sqrt{2} \cdot -0.5\right) \cdot \sqrt{F \cdot \frac{-2}{C}}\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 20
Error	50.1
Cost	13448

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 21
Error	52.9
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -6.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.05 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(\left(A - C\right) - \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	52.9
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 4.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	55.3
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{-135}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(B - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Error	55.9
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -9.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 25
Error	56.5
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq 9 \cdot 10^{-106}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 26
Error	57.0
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -6.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C + 2 \cdot A\right)\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 27
Error	56.6
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -6.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(4 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 28
Error	58.3
Cost	7808

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

Alternative 29
Error	62.6
Cost	7680

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

Alternative 30
Error	62.6
Cost	7680

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

Alternative 31
Error	63.1
Cost	7552

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

Alternative 32
Error	63.0
Cost	7104

\[\frac{1}{\frac{C}{B \cdot \left(-0.5 \cdot \sqrt{\frac{F}{A}}\right)}} \]

Alternative 33
Error	63.0
Cost	7104

\[\frac{\frac{\sqrt{\frac{F}{A}}}{\frac{C}{-0.5}}}{\frac{1}{B}} \]

Alternative 34
Error	63.1
Cost	6976

\[\sqrt{\frac{F}{A}} \cdot \left(B \cdot \frac{-0.5}{C}\right) \]

Reproduce

herbie shell --seed 2023067 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))