ABCF->ab-angle b

Average Accuracy: 18.6% → 40.9%

Time: 1.0min

Precision: binary64

Cost: 34788

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\\ t_1 := t_0 \cdot \frac{-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ t_2 := \frac{t_0 \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ t_3 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ \mathbf{if}\;B \leq -1.95:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_3}}}\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{A + A}}}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_3} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (fma C (* A -8.0) (* B (* B 2.0)))))
        (t_1
         (*
          t_0
          (/ (- (sqrt (* F (- A (hypot B A))))) (fma B B (* -4.0 (* A C))))))
        (t_2 (/ (* t_0 (- (sqrt (* F (+ A A))))) (fma B B (* A (* C -4.0)))))
        (t_3 (* (- A (hypot A B)) F)))
   (if (<= B -1.95)
     (/ 1.0 (/ B (sqrt (* 2.0 t_3))))
     (if (<= B -5.8e-213)
       t_1
       (if (<= B -1.55e-285)
         (/
          (-
           (sqrt
            (*
             2.0
             (fma
              C
              (* F (+ (* B B) (* -8.0 (* A A))))
              (* 2.0 (* A (* B (* B F))))))))
          (- (* B B) (* (* A C) 4.0)))
         (if (<= B 5.1e-287)
           (sqrt (/ F (* -2.0 (/ (* A C) (+ A A)))))
           (if (<= B 1.3e-237)
             t_2
             (if (<= B 6.5e-169)
               (sqrt (/ F (- C)))
               (if (<= B 2.9e-128)
                 t_2
                 (if (<= B 2.5e-59)
                   (sqrt (/ F (- (/ (* (* B B) 0.25) A) C)))
                   (if (<= B 8.6e+49)
                     t_1
                     (* (sqrt t_3) (/ (- (sqrt 2.0)) B)))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = sqrt(fma(C, (A * -8.0), (B * (B * 2.0))));
	double t_1 = t_0 * (-sqrt((F * (A - hypot(B, A)))) / fma(B, B, (-4.0 * (A * C))));
	double t_2 = (t_0 * -sqrt((F * (A + A)))) / fma(B, B, (A * (C * -4.0)));
	double t_3 = (A - hypot(A, B)) * F;
	double tmp;
	if (B <= -1.95) {
		tmp = 1.0 / (B / sqrt((2.0 * t_3)));
	} else if (B <= -5.8e-213) {
		tmp = t_1;
	} else if (B <= -1.55e-285) {
		tmp = -sqrt((2.0 * fma(C, (F * ((B * B) + (-8.0 * (A * A)))), (2.0 * (A * (B * (B * F))))))) / ((B * B) - ((A * C) * 4.0));
	} else if (B <= 5.1e-287) {
		tmp = sqrt((F / (-2.0 * ((A * C) / (A + A)))));
	} else if (B <= 1.3e-237) {
		tmp = t_2;
	} else if (B <= 6.5e-169) {
		tmp = sqrt((F / -C));
	} else if (B <= 2.9e-128) {
		tmp = t_2;
	} else if (B <= 2.5e-59) {
		tmp = sqrt((F / ((((B * B) * 0.25) / A) - C)));
	} else if (B <= 8.6e+49) {
		tmp = t_1;
	} else {
		tmp = sqrt(t_3) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = sqrt(fma(C, Float64(A * -8.0), Float64(B * Float64(B * 2.0))))
	t_1 = Float64(t_0 * Float64(Float64(-sqrt(Float64(F * Float64(A - hypot(B, A))))) / fma(B, B, Float64(-4.0 * Float64(A * C)))))
	t_2 = Float64(Float64(t_0 * Float64(-sqrt(Float64(F * Float64(A + A))))) / fma(B, B, Float64(A * Float64(C * -4.0))))
	t_3 = Float64(Float64(A - hypot(A, B)) * F)
	tmp = 0.0
	if (B <= -1.95)
		tmp = Float64(1.0 / Float64(B / sqrt(Float64(2.0 * t_3))));
	elseif (B <= -5.8e-213)
		tmp = t_1;
	elseif (B <= -1.55e-285)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * fma(C, Float64(F * Float64(Float64(B * B) + Float64(-8.0 * Float64(A * A)))), Float64(2.0 * Float64(A * Float64(B * Float64(B * F)))))))) / Float64(Float64(B * B) - Float64(Float64(A * C) * 4.0)));
	elseif (B <= 5.1e-287)
		tmp = sqrt(Float64(F / Float64(-2.0 * Float64(Float64(A * C) / Float64(A + A)))));
	elseif (B <= 1.3e-237)
		tmp = t_2;
	elseif (B <= 6.5e-169)
		tmp = sqrt(Float64(F / Float64(-C)));
	elseif (B <= 2.9e-128)
		tmp = t_2;
	elseif (B <= 2.5e-59)
		tmp = sqrt(Float64(F / Float64(Float64(Float64(Float64(B * B) * 0.25) / A) - C)));
	elseif (B <= 8.6e+49)
		tmp = t_1;
	else
		tmp = Float64(sqrt(t_3) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(C * N[(A * -8.0), $MachinePrecision] + N[(B * N[(B * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[((-N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * (-N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[B, -1.95], N[(1.0 / N[(B / N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -5.8e-213], t$95$1, If[LessEqual[B, -1.55e-285], N[((-N[Sqrt[N[(2.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-8.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * N[(B * N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.1e-287], N[Sqrt[N[(F / N[(-2.0 * N[(N[(A * C), $MachinePrecision] / N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, 1.3e-237], t$95$2, If[LessEqual[B, 6.5e-169], N[Sqrt[N[(F / (-C)), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, 2.9e-128], t$95$2, If[LessEqual[B, 2.5e-59], N[Sqrt[N[(F / N[(N[(N[(N[(B * B), $MachinePrecision] * 0.25), $MachinePrecision] / A), $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, 8.6e+49], t$95$1, N[(N[Sqrt[t$95$3], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -1.94999999999999996

   1. Initial program 14.5%

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

   2. Simplified 12.4%

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

      [Start] 14.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 1.1%

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

      [Start] 12.4	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 0.6	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 0.9	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      frac-times [=>] 0.9	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right) \cdot \left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]

   4. Simplified 1.2%

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

      [Start] 1.1	\[ \sqrt{\frac{F \cdot \left(\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 1.2	\[ \sqrt{\color{blue}{\frac{F}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}}} \]

   5. Taylor expanded in C around 0 20.2%

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

   6. Simplified 47.5%

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

      [Start] 20.2	\[ \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]
      *-commutative [=>] 20.2	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      +-commutative [=>] 20.2	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{\color{blue}{{A}^{2} + {B}^{2}}}\right) \cdot F} \]
      unpow2 [=>] 20.2	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right) \cdot F} \]
      unpow2 [=>] 20.2	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right) \cdot F} \]
      hypot-def [=>] 47.5	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right) \cdot F} \]

   7. Applied egg-rr 47.6%

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

      [Start] 47.5	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \]
      associate-*l/ [=>] 47.5	\[ \color{blue}{\frac{\sqrt{2} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}}{B}} \]
      clear-num [=>] 47.5	\[ \color{blue}{\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F}}}} \]
      sqrt-unprod [=>] 47.6	\[ \frac{1}{\frac{B}{\color{blue}{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}}} \]

   if -1.94999999999999996 < B < -5.7999999999999999e-213 or 2.5000000000000001e-59 < B < 8.5999999999999998e49

   1. Initial program 28.0%

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

   2. Simplified 33.9%

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

      [Start] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 33.0%

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

      [Start] 33.9	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 37.7	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-prod [=>] 34.9	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--r- [=>] 33.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A - \mathsf{hypot}\left(B, A - C\right)\right) + C\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-commutative [=>] 33.0	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 33.0	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{\left(B \cdot B\right) \cdot 2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 33.0	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{B \cdot \left(B \cdot 2\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Simplified 33.0%

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

      [Start] 33.0	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [<=] 33.0	\[ \frac{-\color{blue}{{\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 33.0	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot {\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [=>] 33.0	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Taylor expanded in C around 0 31.7%

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

   6. Simplified 37.6%

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

      [Start] 31.7	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 31.7	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 31.7	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      hypot-def [=>] 37.6	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Applied egg-rr 36.2%

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

      [Start] 37.6	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      div-inv [=>] 37.3	\[ \color{blue}{\left(-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      distribute-rgt-neg-in [=>] 37.3	\[ \color{blue}{\left(\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 36.2	\[ \color{blue}{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(\left(-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]

   8. Simplified 36.3%

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

      [Start] 36.2	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(\left(-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      *-commutative [=>] 36.2	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \color{blue}{\left(2 \cdot B\right)}\right)} \cdot \left(\left(-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 36.3	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \color{blue}{\frac{\left(-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-rgt-identity [=>] 36.3	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \frac{\color{blue}{-\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 36.3	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \frac{-\sqrt{\color{blue}{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 36.3	\[ \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(2 \cdot B\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)} \]

   if -5.7999999999999999e-213 < B < -1.55e-285

   1. Initial program 19.4%

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

   2. Simplified 19.3%

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

      [Start] 19.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 26.9%

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

   4. Simplified 26.9%

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

      [Start] 26.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - -1 \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 26.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) - \color{blue}{\left(-A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 31.4%

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

   6. Simplified 31.0%

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

      [Start] 31.4	\[ \frac{-\sqrt{2 \cdot \left(2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + \left(F \cdot {B}^{2} + -8 \cdot \left({A}^{2} \cdot F\right)\right) \cdot C\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      +-commutative [=>] 31.4	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(F \cdot {B}^{2} + -8 \cdot \left({A}^{2} \cdot F\right)\right) \cdot C + 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 31.4	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{C \cdot \left(F \cdot {B}^{2} + -8 \cdot \left({A}^{2} \cdot F\right)\right)} + 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 31.4	\[ \frac{-\sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(C, F \cdot {B}^{2} + -8 \cdot \left({A}^{2} \cdot F\right), 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 31.4	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, \color{blue}{{B}^{2} \cdot F} + -8 \cdot \left({A}^{2} \cdot F\right), 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, {B}^{2} \cdot F + \color{blue}{\left(-8 \cdot {A}^{2}\right) \cdot F}, 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-rgt-out [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, \color{blue}{F \cdot \left({B}^{2} + -8 \cdot {A}^{2}\right)}, 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(\color{blue}{B \cdot B} + -8 \cdot {A}^{2}\right), 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \color{blue}{\left(A \cdot A\right)}\right), 2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 31.0	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \color{blue}{\left(B \cdot \left(B \cdot F\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.55e-285 < B < 5.0999999999999998e-287

   1. Initial program 15.3%

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

   2. Simplified 22.7%

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

      [Start] 15.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 7.3%

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

      [Start] 22.7	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 9.3	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 8.9	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      frac-times [=>] 8.0	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right) \cdot \left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]

   4. Simplified 8.3%

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

      [Start] 7.3	\[ \sqrt{\frac{F \cdot \left(\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 8.3	\[ \sqrt{\color{blue}{\frac{F}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}}} \]

   5. Taylor expanded in C around inf 28.8%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{-2 \cdot \frac{A \cdot C}{A - -1 \cdot A}}}} \]

   6. Simplified 28.8%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{-2 \cdot \frac{A \cdot C}{A + A}}}} \]
      Proof

      [Start] 28.8	\[ \sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{A - -1 \cdot A}}} \]
      cancel-sign-sub-inv [=>] 28.8	\[ \sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{\color{blue}{A + \left(--1\right) \cdot A}}}} \]
      metadata-eval [=>] 28.8	\[ \sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{A + \color{blue}{1} \cdot A}}} \]
      *-lft-identity [=>] 28.8	\[ \sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{A + \color{blue}{A}}}} \]

   if 5.0999999999999998e-287 < B < 1.3000000000000001e-237 or 6.5000000000000002e-169 < B < 2.9e-128

   1. Initial program 20.9%

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

   2. Simplified 27.7%

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

      [Start] 20.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 21.4%

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

      [Start] 27.7	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 32.6	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-prod [=>] 23.6	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--r- [=>] 21.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(A - \mathsf{hypot}\left(B, A - C\right)\right) + C\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-commutative [=>] 21.4	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 21.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{\left(B \cdot B\right) \cdot 2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 21.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, \color{blue}{B \cdot \left(B \cdot 2\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Simplified 21.4%

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

      [Start] 21.4	\[ \frac{-\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [<=] 21.4	\[ \frac{-\color{blue}{{\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 21.4	\[ \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot {\left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}^{0.5}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow1/2 [=>] 21.4	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Taylor expanded in C around inf 31.5%

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

   6. Simplified 31.5%

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

      [Start] 31.5	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\left(A - -1 \cdot A\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 31.5	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      cancel-sign-sub-inv [=>] 31.5	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(--1\right) \cdot A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      metadata-eval [=>] 31.5	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{F \cdot \left(A + \color{blue}{1} \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-lft-identity [=>] 31.5	\[ \frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{F \cdot \left(A + \color{blue}{A}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 1.3000000000000001e-237 < B < 6.5000000000000002e-169

   1. Initial program 17.2%

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

   2. Simplified 25.5%

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

      [Start] 17.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 8.9%

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

      [Start] 25.5	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 13.6	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 11.1	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      frac-times [=>] 9.5	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right) \cdot \left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]

   4. Simplified 9.4%

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

      [Start] 8.9	\[ \sqrt{\frac{F \cdot \left(\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 9.4	\[ \sqrt{\color{blue}{\frac{F}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}}} \]

   5. Taylor expanded in A around -inf 37.5%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{-1 \cdot C}}} \]

   6. Simplified 37.5%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{-C}}} \]
      Proof

      [Start] 37.5	\[ \sqrt{\frac{F}{-1 \cdot C}} \]
      mul-1-neg [=>] 37.5	\[ \sqrt{\frac{F}{\color{blue}{-C}}} \]

   if 2.9e-128 < B < 2.5000000000000001e-59

   1. Initial program 24.9%

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

   2. Simplified 31.7%

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

      [Start] 24.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 8.2%

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

      [Start] 31.7	\[ \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 12.7	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 10.0	\[ \color{blue}{\sqrt{\frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]
      frac-times [=>] 9.3	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right) \cdot \left(-\sqrt{F \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, 2 \cdot \left(B \cdot B\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}} \]

   4. Simplified 9.3%

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

      [Start] 8.2	\[ \sqrt{\frac{F \cdot \left(\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 9.3	\[ \sqrt{\color{blue}{\frac{F}{\frac{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}}}} \]

   5. Taylor expanded in A around -inf 34.0%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{-1 \cdot \frac{-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}}{A} + -1 \cdot C}}} \]

   6. Simplified 34.0%

      \[\leadsto \sqrt{\frac{F}{\color{blue}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}} \]
      Proof

      [Start] 34.0	\[ \sqrt{\frac{F}{-1 \cdot \frac{-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}}{A} + -1 \cdot C}} \]
      mul-1-neg [=>] 34.0	\[ \sqrt{\frac{F}{-1 \cdot \frac{-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}}{A} + \color{blue}{\left(-C\right)}}} \]
      unsub-neg [=>] 34.0	\[ \sqrt{\frac{F}{\color{blue}{-1 \cdot \frac{-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}}{A} - C}}} \]
      associate-*r/ [=>] 34.0	\[ \sqrt{\frac{F}{\color{blue}{\frac{-1 \cdot \left(-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}\right)}{A}} - C}} \]
      neg-mul-1 [<=] 34.0	\[ \sqrt{\frac{F}{\frac{\color{blue}{-\left(-0.5 \cdot {B}^{2} - -0.25 \cdot {B}^{2}\right)}}{A} - C}} \]
      distribute-rgt-out-- [=>] 34.0	\[ \sqrt{\frac{F}{\frac{-\color{blue}{{B}^{2} \cdot \left(-0.5 - -0.25\right)}}{A} - C}} \]
      metadata-eval [=>] 34.0	\[ \sqrt{\frac{F}{\frac{-{B}^{2} \cdot \color{blue}{-0.25}}{A} - C}} \]
      distribute-rgt-neg-in [=>] 34.0	\[ \sqrt{\frac{F}{\frac{\color{blue}{{B}^{2} \cdot \left(--0.25\right)}}{A} - C}} \]
      unpow2 [=>] 34.0	\[ \sqrt{\frac{F}{\frac{\color{blue}{\left(B \cdot B\right)} \cdot \left(--0.25\right)}{A} - C}} \]
      metadata-eval [=>] 34.0	\[ \sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot \color{blue}{0.25}}{A} - C}} \]

   if 8.5999999999999998e49 < B

   1. Initial program 9.9%

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

   2. Simplified 9.9%

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

      [Start] 9.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 16.6%

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

   4. Simplified 50.0%

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

      [Start] 16.6	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 16.6	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 16.6	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 16.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]
      +-commutative [=>] 16.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{{A}^{2} + {B}^{2}}}\right)}\right) \]
      unpow2 [=>] 16.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]
      unpow2 [=>] 16.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{A \cdot A + \color{blue}{B \cdot B}}\right)}\right) \]
      hypot-def [=>] 50.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(A, B\right)}\right)}\right) \]

3. Recombined 8 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.95:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}}\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\frac{F}{-2 \cdot \frac{A \cdot C}{A + A}}}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-237}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	41.3%
Cost	34128

\[\begin{array}{l} t_0 := \frac{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ t_1 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ \mathbf{if}\;B \leq -1.85:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_1}}}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-248}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Accuracy	41.0%
Cost	28060

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ t_2 := \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\\ t_3 := \frac{t_2 \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{\mathsf{fma}\left(B, B, t_0\right)}\\ t_4 := \sqrt{\frac{F}{-C}}\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_1}}}\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-141}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.92 \cdot 10^{-286}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-241}:\\ \;\;\;\;-\frac{t_2 \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{-170}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-128}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Accuracy	41.3%
Cost	28060

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ t_3 := \sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{F \cdot \left(A + A\right)}\right)}{\mathsf{fma}\left(B, B, t_0\right)}\\ t_5 := \sqrt{\frac{F}{-C}}\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_2}}}\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-185}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-287}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-241}:\\ \;\;\;\;-\frac{t_3 \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-167}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-128}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Accuracy	40.1%
Cost	27468

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{-C}}\\ t_1 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_1}}}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-286}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;-\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -8, B \cdot \left(B \cdot 2\right)\right)} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.68 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Accuracy	40.3%
Cost	20432

\[\begin{array}{l} t_0 := \left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\\ \mathbf{if}\;B \leq -2 \cdot 10^{-63}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_0}}}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \mathsf{fma}\left(C, F \cdot \left(B \cdot B + -8 \cdot \left(A \cdot A\right)\right), 2 \cdot \left(A \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)\right)}}{B \cdot B - \left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Accuracy	24.3%
Cost	20104

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := A - \mathsf{hypot}\left(A, B\right)\\ t_2 := t_1 \cdot F\\ \mathbf{if}\;F \leq -6.5 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{B}{\sqrt{t_2}}}\\ \mathbf{elif}\;F \leq -9.8 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{t_0}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2 \cdot t_2}}}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(\left(A - -0.5 \cdot \frac{B \cdot B}{A}\right) - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \end{array} \]

Alternative 7
Accuracy	24.3%
Cost	14932

\[\begin{array}{l} t_0 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_1 := A - \mathsf{hypot}\left(A, B\right)\\ t_2 := \frac{1}{\frac{B}{\sqrt{2 \cdot \left(t_1 \cdot F\right)}}}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{+111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{t_0}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-234}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(\left(A - -0.5 \cdot \frac{B \cdot B}{A}\right) - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \end{array} \]

Alternative 8
Accuracy	25.0%
Cost	14096

\[\begin{array}{l} t_0 := \frac{1}{\frac{B}{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right)}}}\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -7 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;F \leq -5.7 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{+27}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{t_1}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-234}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \left(\left(A - -0.5 \cdot \frac{B \cdot B}{A}\right) - C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \end{array} \]

Alternative 9
Accuracy	25.0%
Cost	13968

\[\begin{array}{l} t_0 := \frac{\sqrt{F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;F \leq -8.5 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{+28}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -16\right)}}{t_1}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-234}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \left(\left(A - -0.5 \cdot \frac{B \cdot B}{A}\right) - C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \end{array} \]

Alternative 10
Accuracy	31.5%
Cost	13776

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;A \leq -1.3 \cdot 10^{+204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.65 \cdot 10^{+75}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -7 \cdot 10^{-141}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;A \leq -1.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	29.3%
Cost	8848

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -1.35 \cdot 10^{+203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -2.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.4 \cdot 10^{-211}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;A \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	27.5%
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ t_1 := B \cdot B - \left(A \cdot C\right) \cdot 4\\ \mathbf{if}\;A \leq -6.4 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.42 \cdot 10^{-211}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	27.2%
Cost	7817

\[\begin{array}{l} \mathbf{if}\;A \leq -6.4 \cdot 10^{+107} \lor \neg \left(A \leq -1.6 \cdot 10^{-162}\right):\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right)}\\ \end{array} \]

Alternative 14
Accuracy	21.2%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{F}{\frac{\left(B \cdot B\right) \cdot 0.25}{A} - C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 15
Accuracy	20.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 16
Accuracy	2.0%
Cost	6656

\[\sqrt{\frac{F}{-A}} \]

Alternative 17
Accuracy	20.1%
Cost	6656

\[\sqrt{\frac{F}{-C}} \]

Alternative 18
Accuracy	1.5%
Cost	6592

\[\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023151 
(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))))