ABCF->ab-angle b

Average Error: 18.4% → 39.6%

Time: 1.8min

Precision: binary64

Cost: 55188.00

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := A - \left(t_2 - C\right)\\ t_4 := t_3 \cdot F\\ t_5 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.95 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_4} \cdot \left(-\sqrt{2 \cdot t_5}\right)}{t_5}\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{-286}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-280}:\\ \;\;\;\;-\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{t_1}} \cdot \left(-\sqrt[3]{t_1}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_5 \cdot \left(\left(A + C\right) - t_2\right)\right)\right)}}{-t_5}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_2 - A\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_4}}{B}\\ \end{array} \]

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 (fma A (* C -4.0) (* B B)))
        (t_1 (* (+ C (fma -0.5 (/ (* B B) A) C)) (* 2.0 (* F t_0))))
        (t_2 (hypot B (- A C)))
        (t_3 (- A (- t_2 C)))
        (t_4 (* t_3 F))
        (t_5 (fma B B (* A (* C -4.0)))))
   (if (<= B -1.95e+98)
     (/ (sqrt (* t_3 (* 2.0 F))) B)
     (if (<= B -7.2e-143)
       (/ (* (sqrt t_4) (- (sqrt (* 2.0 t_5)))) t_5)
       (if (<= B -3.4e-286)
         (/ (- (sqrt (* 2.0 (* -4.0 (* (* A C) (* F (+ C C))))))) t_5)
         (if (<= B 7.5e-280)
           (- (/ (sqrt (* t_0 (* (* 2.0 F) (* A 2.0)))) t_0))
           (if (<= B 1.75e-195)
             (/ (* (cbrt (sqrt t_1)) (- (cbrt t_1))) t_0)
             (if (<= B 1.45e-114)
               (/ (sqrt (* 2.0 (* F (* t_5 (- (+ A C) t_2))))) (- t_5))
               (if (<= B 1.5e+141)
                 (/
                  (* (sqrt t_0) (- (sqrt (* (* 2.0 F) (- C (- t_2 A))))))
                  t_0)
                 (/ (- (sqrt (* 2.0 t_4))) 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 = fma(A, (C * -4.0), (B * B));
	double t_1 = (C + fma(-0.5, ((B * B) / A), C)) * (2.0 * (F * t_0));
	double t_2 = hypot(B, (A - C));
	double t_3 = A - (t_2 - C);
	double t_4 = t_3 * F;
	double t_5 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -1.95e+98) {
		tmp = sqrt((t_3 * (2.0 * F))) / B;
	} else if (B <= -7.2e-143) {
		tmp = (sqrt(t_4) * -sqrt((2.0 * t_5))) / t_5;
	} else if (B <= -3.4e-286) {
		tmp = -sqrt((2.0 * (-4.0 * ((A * C) * (F * (C + C)))))) / t_5;
	} else if (B <= 7.5e-280) {
		tmp = -(sqrt((t_0 * ((2.0 * F) * (A * 2.0)))) / t_0);
	} else if (B <= 1.75e-195) {
		tmp = (cbrt(sqrt(t_1)) * -cbrt(t_1)) / t_0;
	} else if (B <= 1.45e-114) {
		tmp = sqrt((2.0 * (F * (t_5 * ((A + C) - t_2))))) / -t_5;
	} else if (B <= 1.5e+141) {
		tmp = (sqrt(t_0) * -sqrt(((2.0 * F) * (C - (t_2 - A))))) / t_0;
	} else {
		tmp = -sqrt((2.0 * t_4)) / 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 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = Float64(Float64(C + fma(-0.5, Float64(Float64(B * B) / A), C)) * Float64(2.0 * Float64(F * t_0)))
	t_2 = hypot(B, Float64(A - C))
	t_3 = Float64(A - Float64(t_2 - C))
	t_4 = Float64(t_3 * F)
	t_5 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -1.95e+98)
		tmp = Float64(sqrt(Float64(t_3 * Float64(2.0 * F))) / B);
	elseif (B <= -7.2e-143)
		tmp = Float64(Float64(sqrt(t_4) * Float64(-sqrt(Float64(2.0 * t_5)))) / t_5);
	elseif (B <= -3.4e-286)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-4.0 * Float64(Float64(A * C) * Float64(F * Float64(C + C))))))) / t_5);
	elseif (B <= 7.5e-280)
		tmp = Float64(-Float64(sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * Float64(A * 2.0)))) / t_0));
	elseif (B <= 1.75e-195)
		tmp = Float64(Float64(cbrt(sqrt(t_1)) * Float64(-cbrt(t_1))) / t_0);
	elseif (B <= 1.45e-114)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(t_5 * Float64(Float64(A + C) - t_2))))) / Float64(-t_5));
	elseif (B <= 1.5e+141)
		tmp = Float64(Float64(sqrt(t_0) * Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(C - Float64(t_2 - A)))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * t_4))) / 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 * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(A - N[(t$95$2 - C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * F), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.95e+98], N[(N[Sqrt[N[(t$95$3 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -7.2e-143], N[(N[(N[Sqrt[t$95$4], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[B, -3.4e-286], N[((-N[Sqrt[N[(2.0 * N[(-4.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[B, 7.5e-280], (-N[(N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * N[(A * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 1.75e-195], N[(N[(N[Power[N[Sqrt[t$95$1], $MachinePrecision], 1/3], $MachinePrecision] * (-N[Power[t$95$1, 1/3], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1.45e-114], N[(N[Sqrt[N[(2.0 * N[(F * N[(t$95$5 * N[(N[(A + C), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$5)), $MachinePrecision], If[LessEqual[B, 1.5e+141], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[(t$95$2 - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -1.95e98

   1. Initial program 5.1

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

   2. Simplified 6.1

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

      [Start] 5.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 13.7

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

   4. Simplified 13.7

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

      [Start] 13.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 13.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 13.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--r- [<=] 13.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 13.8

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

   6. Simplified 16.3

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

      [Start] 13.8	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 13.8	\[ \color{blue}{\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 16.3	\[ \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 16.3	\[ \color{blue}{\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 16.3	\[ \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 16.3	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-*l* [=>] 16.3	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around -inf 54.0

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

   8. Simplified 54.0

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

      [Start] 54.0	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-1 \cdot B} \]
      mul-1-neg [=>] 54.0	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{-B}} \]

   9. Applied egg-rr 54.1

      \[\leadsto \color{blue}{0 + \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}} \]

   10. Simplified 54.0

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

       [Start] 54.1	\[ 0 + \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{B} \]
       +-lft-identity [=>] 54.1	\[ \color{blue}{\frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}} \]
       associate--l+ [=>] 54.0	\[ \frac{\sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{B} \]

   if -1.95e98 < B < -7.1999999999999996e-143

   1. Initial program 30.9

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

   2. Simplified 38.4

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

      [Start] 30.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 39.2

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

   4. Simplified 39.2

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

      [Start] 39.2	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 39.2	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 39.2	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -7.1999999999999996e-143 < B < -3.4000000000000001e-286

   1. Initial program 16.1

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

   2. Simplified 26.5

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

      [Start] 16.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 19.5

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

   4. Simplified 23.8

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

      [Start] 19.5	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot \left(\left(C - -1 \cdot C\right) \cdot F\right)\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \color{blue}{\left(F \cdot \left(C - -1 \cdot C\right)\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      mul-1-neg [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if -3.4000000000000001e-286 < B < 7.4999999999999999e-280

   1. Initial program 17.4

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

   2. Simplified 28.6

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

      [Start] 17.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 24.4

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

   if 7.4999999999999999e-280 < B < 1.75000000000000007e-195

   1. Initial program 19.1

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

   2. Simplified 26.2

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

      [Start] 19.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 19.5

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

   4. Simplified 19.5

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

      [Start] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(C + -0.5 \cdot \frac{\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}}{A}\right) - -1 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\color{blue}{\left(-0.5 \cdot \frac{\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}}{A} + C\right)} - -1 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}}{A} + \left(C - -1 \cdot C\right)\right)} \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*r/ [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\color{blue}{\frac{-0.5 \cdot \left(\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}\right)}{A}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-/l* [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\color{blue}{\frac{-0.5}{\frac{A}{\left({B}^{2} + {C}^{2}\right) - {\left(-1 \cdot C\right)}^{2}}}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\frac{-0.5}{\frac{A}{\color{blue}{{B}^{2} + \left({C}^{2} - {\left(-1 \cdot C\right)}^{2}\right)}}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\frac{-0.5}{\frac{A}{\color{blue}{B \cdot B} + \left({C}^{2} - {\left(-1 \cdot C\right)}^{2}\right)}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\frac{-0.5}{\frac{A}{B \cdot B + \left(\color{blue}{C \cdot C} - {\left(-1 \cdot C\right)}^{2}\right)}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\frac{-0.5}{\frac{A}{B \cdot B + \left(C \cdot C - {\color{blue}{\left(-C\right)}}^{2}\right)}} + \left(C - -1 \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 19.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\frac{-0.5}{\frac{A}{B \cdot B + \left(C \cdot C - {\left(-C\right)}^{2}\right)}} + \left(C - \color{blue}{\left(-C\right)}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 21.5

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{\sqrt{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}} \cdot \sqrt[3]{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   6. Simplified 21.5

      \[\leadsto \frac{-\color{blue}{\sqrt[3]{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)} \cdot \sqrt[3]{\sqrt{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      Proof

      [Start] 21.5

      \[ \frac{-\sqrt[3]{\sqrt{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}} \cdot \sqrt[3]{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

      *-commutative [=>] 21.5

      \[ \frac{-\color{blue}{\sqrt[3]{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \cdot \sqrt[3]{\sqrt{\left(C + \left(C + -0.5 \cdot \frac{\mathsf{fma}\left(B, B, \left(-C\right) \cdot \left(C - C\right)\right)}{A}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 1.75000000000000007e-195 < B < 1.44999999999999998e-114

   1. Initial program 18.3

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

   2. Simplified 27.0

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

      [Start] 18.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 26.6

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

   4. Simplified 19.3

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

      [Start] 26.6	\[ \sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r/ [=>] 26.9	\[ \color{blue}{\frac{\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)} \cdot 1}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-rgt-identity [=>] 26.9	\[ \frac{\color{blue}{\sqrt{2 \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 21.0	\[ \frac{\sqrt{2 \cdot \color{blue}{\left(\left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot F\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-+r- [=>] 19.3	\[ \frac{\sqrt{2 \cdot \left(\left(\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.3	\[ \frac{\sqrt{2 \cdot \left(\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.3	\[ \frac{\sqrt{2 \cdot \left(\left(\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)} \]

   if 1.44999999999999998e-114 < B < 1.4999999999999999e141

   1. Initial program 30.0

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

   2. Simplified 37.6

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

      [Start] 30.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 42.3

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

   4. Simplified 43.0

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

      [Start] 42.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 42.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 42.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--r- [<=] 43.0	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 1.4999999999999999e141 < B

   1. Initial program 1.5

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

   2. Simplified 1.8

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

      [Start] 1.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 4.2

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

   4. Simplified 4.2

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

      [Start] 4.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 4.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 4.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--r- [<=] 4.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 4.2

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

   6. Simplified 7.2

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

      [Start] 4.2	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 4.2	\[ \color{blue}{\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 7.2	\[ \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 7.2	\[ \color{blue}{\frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 7.2	\[ \frac{-\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 7.2	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-*l* [=>] 7.2	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around 0 53.2

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{B}} \]
3. Recombined 8 regimes into one program.

4. Final simplification 39.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.95 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot F} \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}\;B \leq -3.4 \cdot 10^{-286}:\\ \;\;\;\;\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 7.5 \cdot 10^{-280}:\\ \;\;\;\;-\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}} \cdot \left(-\sqrt[3]{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot F\right)}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	39.5%
Cost	34780.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := A - \left(t_1 - C\right)\\ t_3 := t_2 \cdot F\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_3} \cdot \left(-\sqrt{2 \cdot t_4}\right)}{t_4}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-279}:\\ \;\;\;\;-\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{-t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot \left(\left(A + C\right) - t_1\right)\right)\right)}}{-t_4}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_1 - A\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_3}}{B}\\ \end{array} \]

Alternative 2
Error	39.5%
Cost	34120.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := A - \left(t_1 - C\right)\\ t_3 := t_2 \cdot F\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_3} \cdot \left(-\sqrt{2 \cdot t_4}\right)}{t_4}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-280}:\\ \;\;\;\;-\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{-t_0}\\ \mathbf{elif}\;B \leq 1.62 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot \left(\left(A + C\right) - t_1\right)\right)\right)}}{-t_4}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_3}}{\sqrt{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_1 - A\right)\right)}}{\sqrt{B}}}{\sqrt{B}}\\ \end{array} \]

Alternative 3
Error	39.2%
Cost	28248.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := A - \left(t_0 - C\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot F\right)}}{\sqrt{t_2}}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-143}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 8.3 \cdot 10^{-280}:\\ \;\;\;\;-\frac{\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)} \cdot \frac{-1}{t_2}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot \left(\left(A + C\right) - t_0\right)\right)\right)}}{-t_4}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+125}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_0 - A\right)\right)}}{\sqrt{B}}}{\sqrt{B}}\\ \end{array} \]

Alternative 4
Error	39.5%
Cost	28248.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := A - \left(t_0 - C\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot F\right)}}{\sqrt{t_2}}\\ t_4 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.3 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7.8 \cdot 10^{-143}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-284}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-281}:\\ \;\;\;\;-\frac{\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-200}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right) \cdot \left(2 \cdot \left(F \cdot t_2\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot \left(\left(A + C\right) - t_0\right)\right)\right)}}{-t_4}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+121}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_0 - A\right)\right)}}{\sqrt{B}}}{\sqrt{B}}\\ \end{array} \]

Alternative 5
Error	39.1%
Cost	27868.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := A - \left(t_1 - C\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_2 \cdot F\right)}}{\sqrt{t_0}}\\ \mathbf{if}\;B \leq -2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{t_2 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-143}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-279}:\\ \;\;\;\;-\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(C - \left(t_1 - A\right)\right)}}{\sqrt{B}}}{\sqrt{B}}\\ \end{array} \]

Alternative 6
Error	36.8%
Cost	21776.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \left(t_0 - C\right)\right) \cdot F\right)}}{\mathsf{fma}\left(-1, B, 2 \cdot \frac{A}{\frac{B}{C}}\right)}\\ \mathbf{elif}\;B \leq 8.1 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(C + C\right) + \frac{-0.5}{\frac{A}{B \cdot B}}\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{-1}}{\frac{B}{\sqrt{F \cdot \left(C - \left(t_0 - A\right)\right)}}}\\ \end{array} \]

Alternative 7
Error	36.9%
Cost	21264.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \left(t_0 - C\right)\right) \cdot F\right)}}{\mathsf{fma}\left(-1, B, 2 \cdot \frac{A}{\frac{B}{C}}\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-30}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(\left(C \cdot -2\right) \cdot \left(F \cdot -2\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{-1}}{\frac{B}{\sqrt{F \cdot \left(C - \left(t_0 - A\right)\right)}}}\\ \end{array} \]

Alternative 8
Error	36.7%
Cost	20804.00

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A - \left(t_0 - C\right)\right) \cdot F\right)}}{\mathsf{fma}\left(-1, B, 2 \cdot \frac{A}{\frac{B}{C}}\right)}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{-1}}{\frac{B}{\sqrt{F \cdot \left(C - \left(t_0 - A\right)\right)}}}\\ \end{array} \]

Alternative 9
Error	36.7%
Cost	20752.00

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(t_1 - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F \cdot \left(\left(A + C\right) - t_1\right)}}{B}\\ \end{array} \]

Alternative 10
Error	36.6%
Cost	20752.00

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(t_1 - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{-1}}{\frac{B}{\sqrt{F \cdot \left(C - \left(t_1 - A\right)\right)}}}\\ \end{array} \]

Alternative 11
Error	36.7%
Cost	14736.00

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ t_1 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 12
Error	34.7%
Cost	14348.00

\[\begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 1.22 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 13
Error	36.2%
Cost	14344.00

\[\begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ \mathbf{if}\;B \leq -4.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 14
Error	35.9%
Cost	14344.00

\[\begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ \mathbf{if}\;B \leq -8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 15
Error	35.0%
Cost	14216.00

\[\begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ \mathbf{if}\;B \leq -9 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 16
Error	34.8%
Cost	14024.00

\[\begin{array}{l} t_0 := A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -6.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot F\right)}}{B}\\ \end{array} \]

Alternative 17
Error	25.0%
Cost	13828.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\sqrt{\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	23.6%
Cost	13700.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(F \cdot \left(\mathsf{hypot}\left(B, A\right) - A\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	22.1%
Cost	8584.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{-99}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	22.3%
Cost	8584.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -6.3 \cdot 10^{-111}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	19.6%
Cost	8452.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.5 \cdot 10^{-107}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	15.9%
Cost	7940.00

\[\begin{array}{l} \mathbf{if}\;B \leq -8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(4 \cdot \left(C \cdot \left(C \cdot F\right)\right)\right)\right) \cdot -2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 23
Error	4.7%
Cost	6980.00

\[\begin{array}{l} \mathbf{if}\;C \leq -1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{B} \cdot \sqrt{C \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0}\\ \end{array} \]

Alternative 24
Error	14.6%
Cost	6976.00

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

Alternative 25
Error	3.8%
Cost	6464.00

\[\sqrt{0} \]

Reproduce

herbie shell --seed 2023093 
(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))))