ABCF->ab-angle b

Average Error: 52.5 → 39.6

Time: 49.1s

Precision: binary64

Cost: 142412

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := -\frac{\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(t_0 - C\right)\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{elif}\;t_3 \leq 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(A + \left(A - \frac{\left({\left(-A\right)}^{2} - A \cdot A\right) - B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{\frac{C + \left(A - t_0\right)}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{2 \cdot F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (hypot B (- A C)))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3
         (-
          (/
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
           t_2))))
   (if (<= t_3 -2e-187)
     (/ (* (sqrt (* F (- A (- t_0 C)))) (- (sqrt (* 2.0 t_1)))) t_1)
     (if (<= t_3 1e-164)
       (/
        (-
         (sqrt
          (*
           t_1
           (*
            (+ A (- A (* (/ (- (- (pow (- A) 2.0) (* A A)) (* B B)) C) -0.5)))
            (* 2.0 F)))))
        t_1)
       (if (<= t_3 INFINITY)
         (sqrt (/ (+ C (- A t_0)) (/ (fma B B (* C (* A -4.0))) (* 2.0 F))))
         (*
          (sqrt (* F (- C (fma 0.5 (* C (/ C B)) B))))
          (/ (- (sqrt 2.0)) B)))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_3 = -(sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2);
	double tmp;
	if (t_3 <= -2e-187) {
		tmp = (sqrt((F * (A - (t_0 - C)))) * -sqrt((2.0 * t_1))) / t_1;
	} else if (t_3 <= 1e-164) {
		tmp = -sqrt((t_1 * ((A + (A - ((((pow(-A, 2.0) - (A * A)) - (B * B)) / C) * -0.5))) * (2.0 * F)))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((C + (A - t_0)) / (fma(B, B, (C * (A * -4.0))) / (2.0 * F))));
	} else {
		tmp = sqrt((F * (C - fma(0.5, (C * (C / B)), B)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2))
	tmp = 0.0
	if (t_3 <= -2e-187)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - Float64(t_0 - C)))) * Float64(-sqrt(Float64(2.0 * t_1)))) / t_1);
	elseif (t_3 <= 1e-164)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(Float64(A + Float64(A - Float64(Float64(Float64(Float64((Float64(-A) ^ 2.0) - Float64(A * A)) - Float64(B * B)) / C) * -0.5))) * Float64(2.0 * F))))) / t_1);
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(C + Float64(A - t_0)) / Float64(fma(B, B, Float64(C * Float64(A * -4.0))) / Float64(2.0 * F))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(C - fma(0.5, Float64(C * Float64(C / B)), B)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $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] / t$95$2), $MachinePrecision])}, If[LessEqual[t$95$3, -2e-187], N[(N[(N[Sqrt[N[(F * N[(A - N[(t$95$0 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 1e-164], N[((-N[Sqrt[N[(t$95$1 * N[(N[(A + N[(A - N[(N[(N[(N[(N[Power[(-A), 2.0], $MachinePrecision] - N[(A * A), $MachinePrecision]), $MachinePrecision] - N[(B * B), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(F * N[(C - N[(0.5 * N[(C * N[(C / B), $MachinePrecision]), $MachinePrecision] + B), $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))) < -2e-187

   1. Initial program 37.9

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

   2. Simplified 32.6

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

      [Start] 37.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 22.2

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

   4. Simplified 22.2

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

      [Start] 22.2	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 22.2	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 22.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -2e-187 < (/.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))) < 9.99999999999999962e-165

   1. Initial program 59.4

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

   2. Simplified 55.8

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

      [Start] 59.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 45.2

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

   4. Simplified 45.2

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

      [Start] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + -0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right) - -1 \cdot A\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\color{blue}{\left(A + \left(-0.5 \cdot \frac{\left({B}^{2} + {A}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C} - -1 \cdot A\right)\right)} \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(-0.5 \cdot \frac{\color{blue}{{B}^{2} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}}{C} - -1 \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(-0.5 \cdot \frac{\color{blue}{B \cdot B} + \left({A}^{2} - {\left(-1 \cdot A\right)}^{2}\right)}{C} - -1 \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(-0.5 \cdot \frac{B \cdot B + \left(\color{blue}{A \cdot A} - {\left(-1 \cdot A\right)}^{2}\right)}{C} - -1 \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(-0.5 \cdot \frac{B \cdot B + \left(A \cdot A - {\color{blue}{\left(-A\right)}}^{2}\right)}{C} - -1 \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 45.2	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(-0.5 \cdot \frac{B \cdot B + \left(A \cdot A - {\left(-A\right)}^{2}\right)}{C} - \color{blue}{\left(-A\right)}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 9.99999999999999962e-165 < (/.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 41.4

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

   2. Simplified 27.8

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

      [Start] 41.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      *-inverses [=>] 64.0	\[ \color{blue}{1} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      associate-*r* [=>] 64.0	\[ 1 \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{\left(A \cdot C\right) \cdot -4}}\right)} \]

   5. Applied egg-rr 40.6

      \[\leadsto 1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\right)} - 1\right)} \]

   6. Simplified 21.8

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

      [Start] 40.6	\[ 1 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\right)} - 1\right) \]
      expm1-def [=>] 23.6	\[ 1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\right)\right)} \]
      expm1-log1p [=>] 22.2	\[ 1 \cdot \color{blue}{\sqrt{\frac{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      associate-/l* [=>] 21.8	\[ 1 \cdot \sqrt{\color{blue}{\frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{2 \cdot F}}}} \]
      +-commutative [=>] 21.8	\[ 1 \cdot \sqrt{\frac{\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{2 \cdot F}}} \]
      associate-+r- [<=] 21.8	\[ 1 \cdot \sqrt{\frac{\color{blue}{C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{2 \cdot F}}} \]

   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 B around inf 64.0

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

   4. Taylor expanded in A around 0 56.4

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

   5. Simplified 55.0

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

      [Start] 56.4	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}\right) \]
      mul-1-neg [=>] 56.4	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}} \]
      distribute-rgt-neg-in [=>] 56.4	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \left(0.5 \cdot \frac{{C}^{2}}{B} + B\right)\right)}\right)} \]
      fma-def [=>] 56.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \color{blue}{\mathsf{fma}\left(0.5, \frac{{C}^{2}}{B}, B\right)}\right)}\right) \]
      unpow2 [=>] 56.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, \frac{\color{blue}{C \cdot C}}{B}, B\right)\right)}\right) \]
      associate-/l* [=>] 55.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, \color{blue}{\frac{C}{\frac{B}{C}}}, B\right)\right)}\right) \]
      associate-/r/ [=>] 55.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, \color{blue}{\frac{C}{B} \cdot C}, B\right)\right)}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 39.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(A - \frac{\left({\left(-A\right)}^{2} - A \cdot A\right) - B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\sqrt{\frac{C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{2 \cdot F}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	43.6
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -3 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(t_0 - C\right)\right)} \cdot \left(B \cdot \sqrt{2}\right)}{t_1}\\ \mathbf{elif}\;B \leq -4.9 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	45.9
Cost	27468

\[\begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \left(B \cdot t_0\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{t_0}{B}\\ \end{array} \]

Alternative 3
Error	45.0
Cost	27468

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}\\ t_2 := -\sqrt{2}\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{t_1 \cdot \left(B \cdot \sqrt{2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{t_1 \cdot \left(B \cdot t_2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{t_2}{B}\\ \end{array} \]

Alternative 4
Error	46.3
Cost	21964

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq 7.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	48.0
Cost	21140

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ t_2 := A \cdot \left(C \cdot -4\right)\\ t_3 := \mathsf{fma}\left(B, B, t_2\right)\\ t_4 := F \cdot \left(C \cdot \left(A + A\right)\right)\\ t_5 := \frac{\sqrt{t_4} \cdot \left(-\sqrt{A \cdot -8}\right)}{t_3}\\ \mathbf{if}\;A \leq -6 \cdot 10^{+152}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;A \leq -7 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(A \cdot -8\right)}}{t_3}\\ \mathbf{elif}\;A \leq -9 \cdot 10^{-211}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -6.5 \cdot 10^{-248}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 6
Error	47.0
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	48.3
Cost	20488

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, A \cdot \frac{A}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Error	48.3
Cost	20488

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	47.2
Cost	20488

\[\begin{array}{l} \mathbf{if}\;B \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, C \cdot \frac{C}{B}, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 10
Error	49.3
Cost	15368

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;C \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 11
Error	47.8
Cost	15368

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -36000000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 12
Error	51.0
Cost	14344

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.55 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.55 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \left(C \cdot \left(A + A\right)\right)\right) \cdot \left(A \cdot -8\right)}}{\mathsf{fma}\left(B, B, t_1\right)}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-219}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 13
Error	51.5
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -8.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2.05 \cdot 10^{-229}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;A \leq 1.45 \cdot 10^{-213}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 14
Error	51.2
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.05 \cdot 10^{-215}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 15
Error	51.3
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -4.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-188}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(\left(A - C\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 6.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 16
Error	54.0
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 5.8 \cdot 10^{-73}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	55.3
Cost	7680

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

Alternative 18
Error	55.7
Cost	7616

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

Alternative 19
Error	62.8
Cost	7488

\[\frac{-1}{B} \cdot \frac{\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot \left(C + C\right)\right)\right)\right)}}{B} \]

Alternative 20
Error	63.0
Cost	7424

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

Alternative 21
Error	62.9
Cost	7296

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

Reproduce

herbie shell --seed 2023039 
(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))))