ABCF->ab-angle b

Average Error: 80.92% → 60.82%

Time: 51.6s

Precision: binary64

Cost: 154956

Math

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

↓

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := {B}^{2} + C \cdot \left(A \cdot -4\right)\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F} \cdot \left(C \cdot \sqrt{A \cdot -8}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \left(-\sqrt{2}\right)}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \end{array} \]

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 (* A (* C -4.0)))
        (t_1 (fma B B t_0))
        (t_2 (+ (pow B 2.0) (* C (* A -4.0))))
        (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-204)
     (/
      (* (sqrt (* F (- A (- (hypot B (- A C)) C)))) (- (sqrt (* 2.0 t_1))))
      t_1)
     (if (<= t_3 2e+138)
       (/
        (-
         (sqrt (* t_1 (* (fma -0.5 (/ (* B B) (- A C)) (* 2.0 C)) (* 2.0 F)))))
        t_1)
       (if (<= t_3 INFINITY)
         (* (sqrt 2.0) (/ (* (sqrt F) (* C (sqrt (* A -8.0)))) t_1))
         (/
          (* (sqrt (* F (- C (hypot B C)))) (- (sqrt 2.0)))
          (hypot B (sqrt t_0))))))))

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 * -4.0);
	double t_1 = fma(B, B, t_0);
	double t_2 = pow(B, 2.0) + (C * (A * -4.0));
	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-204) {
		tmp = (sqrt((F * (A - (hypot(B, (A - C)) - C)))) * -sqrt((2.0 * t_1))) / t_1;
	} else if (t_3 <= 2e+138) {
		tmp = -sqrt((t_1 * (fma(-0.5, ((B * B) / (A - C)), (2.0 * C)) * (2.0 * F)))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * ((sqrt(F) * (C * sqrt((A * -8.0)))) / t_1);
	} else {
		tmp = (sqrt((F * (C - hypot(B, C)))) * -sqrt(2.0)) / hypot(B, sqrt(t_0));
	}
	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 * -4.0))
	t_1 = fma(B, B, t_0)
	t_2 = Float64((B ^ 2.0) + Float64(C * Float64(A * -4.0)))
	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-204)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - Float64(hypot(B, Float64(A - C)) - C)))) * Float64(-sqrt(Float64(2.0 * t_1)))) / t_1);
	elseif (t_3 <= 2e+138)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(fma(-0.5, Float64(Float64(B * B) / Float64(A - C)), Float64(2.0 * C)) * Float64(2.0 * F))))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(2.0) * Float64(Float64(sqrt(F) * Float64(C * sqrt(Float64(A * -8.0)))) / t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C - hypot(B, C)))) * Float64(-sqrt(2.0))) / hypot(B, sqrt(t_0)));
	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 * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $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-204], N[(N[(N[Sqrt[N[(F * N[(A - N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - 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, 2e+138], N[((-N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[(B * B), $MachinePrecision] / N[(A - C), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Sqrt[F], $MachinePrecision] * N[(C * N[Sqrt[N[(A * -8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision] / N[Sqrt[B ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $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-204

   1. Initial program 57.18

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

   2. Simplified 48.11

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

      [Start] 57.18	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 33.65

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

   4. Simplified 33.65

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

      [Start] 33.65	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 33.65	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 33.65	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -2e-204 < (/.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))) < 2.0000000000000001e138

   1. Initial program 80.02

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

   2. Simplified 75.7

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

      [Start] 80.02	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 65.13

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

   4. Simplified 65.13

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

      [Start] 65.13	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(-0.5 \cdot \frac{{B}^{2}}{A - C} + 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      fma-def [=>] 65.13	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A - C}, 2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 65.13	\[ \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A - C}, 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 2.0000000000000001e138 < (/.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 85.48

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

   2. Simplified 57.84

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

      [Start] 85.48	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 56.58

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

   4. Simplified 72.91

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

      [Start] 56.58	\[ \sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r/ [=>] 55.94	\[ \color{blue}{\frac{\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot 1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-rgt-identity [=>] 55.94	\[ \frac{\color{blue}{\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 72.91	\[ \frac{\sqrt{2 \cdot \color{blue}{\left(\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot F\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+r- [=>] 72.91	\[ \frac{\sqrt{2 \cdot \left(\left(\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 72.91	\[ \frac{\sqrt{2 \cdot \left(\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 72.91	\[ \frac{\sqrt{2 \cdot \left(\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)} \]

   5. Taylor expanded in C around -inf 92.39

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

   6. Simplified 92.38

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

      [Start] 92.39	\[ \frac{\sqrt{2 \cdot \left(\left(-8 \cdot \left(A \cdot {C}^{2}\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 92.38	\[ \frac{\sqrt{2 \cdot \left(\color{blue}{\left(\left(-8 \cdot A\right) \cdot {C}^{2}\right)} \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 92.38	\[ \frac{\sqrt{2 \cdot \left(\left(\left(-8 \cdot A\right) \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Applied egg-rr 55.54

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{F} \cdot \left(C \cdot \sqrt{-8 \cdot A}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   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 100

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

   2. Simplified 98.66

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

      [Start] 100	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 99.25

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

   4. Simplified 99.17

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

      [Start] 99.25	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 99.25	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{\color{blue}{\left(A \cdot C\right) \cdot -4}}\right) \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-commutative [=>] 99.25	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{\left(A \cdot C\right) \cdot -4}\right) \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 99.17	\[ \frac{-\mathsf{hypot}\left(B, \sqrt{\left(A \cdot C\right) \cdot -4}\right) \cdot \sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 81.82

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

   6. Simplified 81.25

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

      [Start] 81.82	\[ \frac{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)} \cdot \frac{-\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)} \]
      *-inverses [=>] 81.24	\[ \color{blue}{1} \cdot \frac{-\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)} \]
      distribute-frac-neg [=>] 81.24	\[ 1 \cdot \color{blue}{\left(-\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)}\right)} \]
      distribute-rgt-neg-in [<=] 81.24	\[ \color{blue}{-1 \cdot \frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)}} \]
      associate-*r/ [=>] 81.24	\[ -\color{blue}{\frac{1 \cdot \sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right)}} \]

   7. Taylor expanded in A around 0 99.2

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

   8. Simplified 81.94

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

      [Start] 99.2	\[ -\frac{\sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \sqrt{2}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      *-commutative [=>] 99.2	\[ -\frac{\color{blue}{\sqrt{2} \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      *-commutative [=>] 99.2	\[ -\frac{\sqrt{2} \cdot \sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      unpow2 [=>] 99.2	\[ -\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      unpow2 [=>] 99.2	\[ -\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]
      hypot-def [=>] 81.94	\[ -\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \color{blue}{\mathsf{hypot}\left(B, C\right)}\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 60.82

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F} \cdot \left(C \cdot \sqrt{A \cdot -8}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \left(-\sqrt{2}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	67.49%
Cost	33808

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\\ t_3 := \sqrt{2 \cdot \left(F \cdot t_2\right)}\\ \mathbf{if}\;A \leq -2.02 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot t_2\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-t_3}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 9.6 \cdot 10^{+147}:\\ \;\;\;\;-\frac{t_3}{\mathsf{hypot}\left(B, \sqrt{C \cdot -4} \cdot \sqrt{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 2
Error	66.34%
Cost	27588

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;A \leq -8.3 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot t_1\right)}}{t_2}\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_1}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 3
Error	65.93%
Cost	27588

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ \mathbf{if}\;A \leq -1.8 \cdot 10^{-187}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(A - \left(t_0 - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq 7.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \left(t_0 - A\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 4
Error	66.49%
Cost	27588

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot t_2\right)}}{t_1}\\ \mathbf{elif}\;A \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_2\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 5
Error	67.64%
Cost	27144

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;A \leq -7 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 3.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}{\mathsf{hypot}\left(B, \sqrt{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 6
Error	73.09%
Cost	21832

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{hypot}\left(B, \sqrt{t_1}\right)\\ \mathbf{if}\;B \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{t_3}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(t_0 - B \cdot B\right)\right) \cdot \left(\left(\sqrt{{\left(A - C\right)}^{2} + B \cdot B} - C\right) - A\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-242}:\\ \;\;\;\;\frac{-1}{C} \cdot \frac{\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot -4}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-272}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+140}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot F\right) \cdot -2}}{t_3}\\ \end{array} \]

Alternative 7
Error	75.68%
Cost	20884

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;B \leq -2.85 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{t_1}\\ \mathbf{elif}\;B \leq -6.7 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{C} \cdot \frac{\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot -4}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{t_0 + B \cdot B}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+141}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot F\right) \cdot -2}}{t_1}\\ \end{array} \]

Alternative 8
Error	75.04%
Cost	20884

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;B \leq -9.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{t_1}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{C} \cdot \frac{\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot -4}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+141}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot F\right) \cdot -2}}{t_1}\\ \end{array} \]

Alternative 9
Error	74.99%
Cost	20884

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, t_0\right)\\ t_2 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ \mathbf{if}\;B \leq -2.85 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \left(B + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{-1}{C} \cdot \frac{\sqrt{t_1 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot -4}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{-t_1}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+141}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot F\right) \cdot -2}}{t_2}\\ \end{array} \]

Alternative 10
Error	77.16%
Cost	20560

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{hypot}\left(B, \sqrt{t_0}\right)\\ t_2 := \mathsf{fma}\left(B, B, t_0\right)\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{-1}{C} \cdot \frac{\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot -4}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{t_0 + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot F\right) \cdot -2}}{t_1}\\ \end{array} \]

Alternative 11
Error	76.36%
Cost	15368

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -6 \cdot 10^{+150}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(\left(\left(A \cdot C\right) \cdot \left(A \cdot F\right)\right) \cdot 8\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 3.9 \cdot 10^{-152}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + A \cdot A}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 12
Error	78.05%
Cost	14984

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(-8 \cdot \left(A \cdot \left(A \cdot C\right)\right)\right) - F \cdot \left(\left(B \cdot \left(B \cdot A\right)\right) \cdot -2\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 3 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 13
Error	83.47%
Cost	8585

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.6 \cdot 10^{-18} \lor \neg \left(B \leq 2.35 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A - \left(B - C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	79.74%
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -3 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 15
Error	80.34%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -9.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot -2\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 16
Error	86%
Cost	7680

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

Alternative 17
Error	90.14%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;B \leq -1.75 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot -16\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 18
Error	95.14%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{-290}:\\ \;\;\;\;2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 19
Error	95.33%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;A \leq -7.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{A \cdot F}}{\frac{B}{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 20
Error	96.93%
Cost	6848

\[2 \cdot \frac{\sqrt{C \cdot F}}{B} \]

Reproduce

herbie shell --seed 2023089 
(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))))