ABCF->ab-angle b

Average Error: 52.5 → 38.9

Time: 55.3s

Precision: binary64

Cost: 155788

Math

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

↓

\[\begin{array}{l} t_0 := A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\\ t_1 := -\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_0} \cdot \left(-\sqrt{2 \cdot t_2}\right)}{t_2}\\ \mathbf{elif}\;t_1 \leq 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B}{\frac{A - C}{B}}\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot t_0} \cdot \left(-\sqrt{2 \cdot F}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (+ A (- C (hypot B (- A C)))))
        (t_1
         (-
          (/
           (sqrt
            (*
             (- (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))) (+ A C))
             (* 2.0 (* F (- (* (* 4.0 A) C) (pow B 2.0))))))
           (+ (pow B 2.0) (* C (* A -4.0))))))
        (t_2 (fma B B (* A (* C -4.0)))))
   (if (<= t_1 -5e-201)
     (/ (* (sqrt (* F t_0)) (- (sqrt (* 2.0 t_2)))) t_2)
     (if (<= t_1 1e-55)
       (-
        (/
         (sqrt
          (* 2.0 (* t_2 (* F (- C (fma -1.0 C (* 0.5 (/ B (/ (- A C) B)))))))))
         t_2))
       (if (<= t_1 INFINITY)
         (/ (* (sqrt (* t_2 t_0)) (- (sqrt (* 2.0 F)))) t_2)
         (* (sqrt (* F (- A (hypot B A)))) (/ (- (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 = A + (C - hypot(B, (A - C)));
	double t_1 = -(sqrt(((sqrt((pow(B, 2.0) + pow((A - C), 2.0))) - (A + C)) * (2.0 * (F * (((4.0 * A) * C) - pow(B, 2.0)))))) / (pow(B, 2.0) + (C * (A * -4.0))));
	double t_2 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (t_1 <= -5e-201) {
		tmp = (sqrt((F * t_0)) * -sqrt((2.0 * t_2))) / t_2;
	} else if (t_1 <= 1e-55) {
		tmp = -(sqrt((2.0 * (t_2 * (F * (C - fma(-1.0, C, (0.5 * (B / ((A - C) / B))))))))) / t_2);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((t_2 * t_0)) * -sqrt((2.0 * F))) / t_2;
	} else {
		tmp = sqrt((F * (A - hypot(B, A)))) * (-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 = Float64(A + Float64(C - hypot(B, Float64(A - C))))
	t_1 = Float64(-Float64(sqrt(Float64(Float64(sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)) * Float64(2.0 * Float64(F * Float64(Float64(Float64(4.0 * A) * C) - (B ^ 2.0)))))) / Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0)))))
	t_2 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_1 <= -5e-201)
		tmp = Float64(Float64(sqrt(Float64(F * t_0)) * Float64(-sqrt(Float64(2.0 * t_2)))) / t_2);
	elseif (t_1 <= 1e-55)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(C - fma(-1.0, C, Float64(0.5 * Float64(B / Float64(Float64(A - C) / B))))))))) / t_2));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(t_2 * t_0)) * Float64(-sqrt(Float64(2.0 * F)))) / t_2);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B, A)))) * 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[(A + N[(C - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(N[Sqrt[N[(N[(N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B, 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 * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-201], N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 1e-55], (-N[(N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(C - N[(-1.0 * C + N[(0.5 * N[(B / N[(N[(A - C), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(t$95$2 * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $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))) < -4.9999999999999999e-201

   1. Initial program 38.1

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

   2. Simplified 32.0

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

      [Start] 38.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 21.5

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

   4. Simplified 21.6

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

      [Start] 21.5	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 21.5	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 21.6	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 21.6	\[ \frac{-\sqrt{F \cdot \left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -4.9999999999999999e-201 < (/.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))) < 9.99999999999999995e-56

   1. Initial program 57.6

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

   2. Simplified 54.5

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

      [Start] 57.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 43.5

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

   4. Simplified 43.5

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

      [Start] 43.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(-1 \cdot C + 0.5 \cdot \frac{{B}^{2}}{A - C}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 43.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \color{blue}{\mathsf{fma}\left(-1, C, 0.5 \cdot \frac{{B}^{2}}{A - C}\right)}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 43.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{\color{blue}{B \cdot B}}{A - C}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-/l* [=>] 43.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \color{blue}{\frac{B}{\frac{A - C}{B}}}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 9.99999999999999995e-56 < (/.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 42.7

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

   2. Simplified 30.3

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

      [Start] 42.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 30.1

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

   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

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

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

   4. Simplified 63.7

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

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

   5. Taylor expanded in C around 0 63.6

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

   6. Simplified 52.7

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

      [Start] 63.6	\[ -1 \cdot \left(\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B}\right) \]
      associate-*r* [=>] 63.6	\[ \color{blue}{\left(-1 \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \cdot \frac{\sqrt{2}}{B}} \]
      *-commutative [=>] 63.6	\[ \left(-1 \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 63.6	\[ \left(-1 \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right) \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 63.6	\[ \left(-1 \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right) \cdot \frac{\sqrt{2}}{B} \]
      neg-mul-1 [<=] 63.6	\[ \color{blue}{\left(-\sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      *-commutative [=>] 63.6	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\right)} \]
      unpow2 [<=] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2}} + A \cdot A}\right)}\right) \]
      unpow2 [<=] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + \color{blue}{{A}^{2}}}\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 38.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -5 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 10^{-55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B}{\frac{A - C}{B}}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	42.9
Cost	34252

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_0}\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-209}:\\ \;\;\;\;-\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{+148}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	45.2
Cost	28556

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;C \leq -1.8 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{F \cdot \left(2 \cdot C\right)}\right)}{t_0}\\ \mathbf{elif}\;C \leq 4.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(F \cdot -2\right) \cdot \left(0.5 \cdot \frac{B \cdot B - \left({\left(-A\right)}^{2} - A \cdot A\right)}{C} - \left(A + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 3
Error	43.2
Cost	28112

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := t_0 \cdot \left(\left(\left(t_1 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)\\ t_3 := \frac{1}{-t_0}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - t_1\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-184}:\\ \;\;\;\;-\frac{{t_2}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;\left(\sqrt{2 \cdot t_0} \cdot \sqrt{F \cdot \left(2 \cdot C\right)}\right) \cdot t_3\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;\sqrt{t_2} \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	43.2
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - t_0\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-97}:\\ \;\;\;\;\left(\sqrt{2 \cdot t_1} \cdot \sqrt{F \cdot \left(2 \cdot C\right)}\right) \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	43.2
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - t_0\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(\left(t_0 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;\left(\sqrt{2 \cdot t_1} \cdot \sqrt{F \cdot \left(2 \cdot C\right)}\right) \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	43.2
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - t_0\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7.1 \cdot 10^{-186}:\\ \;\;\;\;-\frac{{\left(t_1 \cdot \left(\left(\left(t_0 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)\right)}^{0.5}}{t_1}\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{-97}:\\ \;\;\;\;\left(\sqrt{2 \cdot t_1} \cdot \sqrt{F \cdot \left(2 \cdot C\right)}\right) \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + \left(A - t_0\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	48.3
Cost	27268

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;C \leq -7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_1} \cdot \left(-\sqrt{F \cdot \left(2 \cdot C\right)}\right)}{t_1}\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \end{array} \]

Alternative 8
Error	45.5
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-92}:\\ \;\;\;\;t_1 \cdot \frac{B \cdot \sqrt{2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	45.6
Cost	20612

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \mathbf{if}\;B \leq -7.1 \cdot 10^{-96}:\\ \;\;\;\;t_0 \cdot \frac{B \cdot \sqrt{2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 10
Error	45.6
Cost	20168

\[\begin{array}{l} \mathbf{if}\;B \leq -6.4 \cdot 10^{-92}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\\ \end{array} \]

Alternative 11
Error	45.7
Cost	20168

\[\begin{array}{l} t_0 := F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\\ \mathbf{if}\;B \leq -8.4 \cdot 10^{-92}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot t_0}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 12
Error	46.3
Cost	15500

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{-96}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{+61}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A - \left(\sqrt{B \cdot B + A \cdot A} - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 13
Error	46.7
Cost	14984

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := 4 \cdot \left(A \cdot C\right) - B \cdot B\\ \mathbf{if}\;A \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(\left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right) \cdot t_1\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 14
Error	46.6
Cost	14472

\[\begin{array}{l} \mathbf{if}\;B \leq -2.9 \cdot 10^{-96}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 15
Error	47.5
Cost	14344

\[\begin{array}{l} \mathbf{if}\;B \leq -1.22 \cdot 10^{-93}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 16
Error	48.0
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -8.5 \cdot 10^{-93}:\\ \;\;\;\;B \cdot \left(\sqrt{B \cdot F} \cdot \frac{\sqrt{2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\right)\\ \mathbf{elif}\;B \leq 1.46 \cdot 10^{-92}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(C \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 17
Error	47.5
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-92}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(C \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 18
Error	50.8
Cost	14088

\[\begin{array}{l} t_0 := F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{\left(-C\right) \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-279}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(\left(A - C\right) - C\right) - A\right) \cdot t_0\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 19
Error	48.8
Cost	14020

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;B \cdot \left(\sqrt{B \cdot F} \cdot \frac{\sqrt{2}}{t_0}\right)\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-123}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 20
Error	50.1
Cost	13704

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 21
Error	55.0
Cost	8584

\[\begin{array}{l} t_0 := F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(B + C\right)\right) \cdot t_0\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot t_0\right)}}{t_1}\\ \end{array} \]

Alternative 22
Error	56.1
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.15 \cdot 10^{-37}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	59.3
Cost	8320

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

Alternative 24
Error	61.8
Cost	7808

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

Alternative 25
Error	62.6
Cost	7680

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

Alternative 26
Error	63.0
Cost	7296

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

Reproduce

herbie shell --seed 2023187 
(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))))