ABCF->ab-angle b

Average Error: 52.7 → 42.8

Time: 55.6s

Precision: binary64

Cost: 21528

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{t_0 \cdot \left(\left(A + A\right) \cdot \left(F \cdot 2\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-79}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-264}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1450:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{F \cdot \left(-0.5 \cdot \left(B \cdot B\right)\right)}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (/ (- (sqrt (* t_0 (* (+ A A) (* F 2.0))))) t_0))
        (t_2 (/ (sqrt 2.0) B)))
   (if (<= B -4.8e+139)
     (* (sqrt (* F (- A (hypot B A)))) t_2)
     (if (<= B -1.2e-79)
       (* t_2 (sqrt (* F (/ (- (* B B)) (+ A (hypot A B))))))
       (if (<= B -8e-223)
         (/ (* (- (sqrt 2.0)) (sqrt (* F (/ (* B -0.5) (/ A B))))) B)
         (if (<= B -1.75e-265)
           t_1
           (if (<= B 9.5e-264)
             (* C (/ (sqrt (* A (* F -16.0))) (fma B B (* C (* A -4.0)))))
             (if (<= B 1.15e-120)
               t_1
               (if (<= B 1450.0)
                 (* t_2 (/ (- (sqrt (* F (* -0.5 (* B B))))) (sqrt A)))
                 (*
                  t_2
                  (- (sqrt (* F (/ 1.0 (/ 1.0 (- A (hypot A 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(B, B, (A * (C * -4.0)));
	double t_1 = -sqrt((t_0 * ((A + A) * (F * 2.0)))) / t_0;
	double t_2 = sqrt(2.0) / B;
	double tmp;
	if (B <= -4.8e+139) {
		tmp = sqrt((F * (A - hypot(B, A)))) * t_2;
	} else if (B <= -1.2e-79) {
		tmp = t_2 * sqrt((F * (-(B * B) / (A + hypot(A, B)))));
	} else if (B <= -8e-223) {
		tmp = (-sqrt(2.0) * sqrt((F * ((B * -0.5) / (A / B))))) / B;
	} else if (B <= -1.75e-265) {
		tmp = t_1;
	} else if (B <= 9.5e-264) {
		tmp = C * (sqrt((A * (F * -16.0))) / fma(B, B, (C * (A * -4.0))));
	} else if (B <= 1.15e-120) {
		tmp = t_1;
	} else if (B <= 1450.0) {
		tmp = t_2 * (-sqrt((F * (-0.5 * (B * B)))) / sqrt(A));
	} else {
		tmp = t_2 * -sqrt((F * (1.0 / (1.0 / (A - hypot(A, 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(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(A + A) * Float64(F * 2.0))))) / t_0)
	t_2 = Float64(sqrt(2.0) / B)
	tmp = 0.0
	if (B <= -4.8e+139)
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B, A)))) * t_2);
	elseif (B <= -1.2e-79)
		tmp = Float64(t_2 * sqrt(Float64(F * Float64(Float64(-Float64(B * B)) / Float64(A + hypot(A, B))))));
	elseif (B <= -8e-223)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(B * -0.5) / Float64(A / B))))) / B);
	elseif (B <= -1.75e-265)
		tmp = t_1;
	elseif (B <= 9.5e-264)
		tmp = Float64(C * Float64(sqrt(Float64(A * Float64(F * -16.0))) / fma(B, B, Float64(C * Float64(A * -4.0)))));
	elseif (B <= 1.15e-120)
		tmp = t_1;
	elseif (B <= 1450.0)
		tmp = Float64(t_2 * Float64(Float64(-sqrt(Float64(F * Float64(-0.5 * Float64(B * B))))) / sqrt(A)));
	else
		tmp = Float64(t_2 * Float64(-sqrt(Float64(F * Float64(1.0 / Float64(1.0 / Float64(A - hypot(A, 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[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(t$95$0 * N[(N[(A + A), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -4.8e+139], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[B, -1.2e-79], N[(t$95$2 * N[Sqrt[N[(F * N[((-N[(B * B), $MachinePrecision]) / N[(A + N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -8e-223], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(B * -0.5), $MachinePrecision] / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, -1.75e-265], t$95$1, If[LessEqual[B, 9.5e-264], N[(C * N[(N[Sqrt[N[(A * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.15e-120], t$95$1, If[LessEqual[B, 1450.0], N[(t$95$2 * N[((-N[Sqrt[N[(F * N[(-0.5 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * (-N[Sqrt[N[(F * N[(1.0 / N[(1.0 / N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.80000000000000016e139

   1. Initial program 63.4

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

   2. Simplified 63.2

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

      [Start] 63.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 62.9

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

   4. Simplified 62.9

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

      [Start] 62.9	\[ \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      /-rgt-identity [=>] 62.9	\[ \color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 62.9	\[ \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \frac{\sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Taylor expanded in C around 0 61.5

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

   6. Simplified 61.5

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

      [Start] 61.5	\[ \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]
      *-commutative [=>] 61.5	\[ \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 61.5	\[ \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 61.5	\[ \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \cdot \frac{\sqrt{2}}{B} \]

   7. Applied egg-rr 35.2

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

   8. Simplified 32.7

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

      [Start] 35.2	\[ \sqrt{F \cdot \left(A - \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(B, A\right)\right)} - 1\right)\right)} \cdot \frac{\sqrt{2}}{B} \]
      expm1-def [=>] 35.2	\[ \sqrt{F \cdot \left(A - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(B, A\right)\right)\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      expm1-log1p [=>] 32.7	\[ \sqrt{F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]

   if -4.80000000000000016e139 < B < -1.20000000000000003e-79

   1. Initial program 43.5

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

   2. Simplified 39.0

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

      [Start] 43.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 63.3

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

   4. Simplified 63.3

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

      [Start] 63.3	\[ \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      /-rgt-identity [=>] 63.3	\[ \color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate--l+ [=>] 63.3	\[ \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \frac{\sqrt{\color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Taylor expanded in C around 0 41.5

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

   6. Simplified 41.5

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

      [Start] 41.5	\[ \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B} \]
      *-commutative [=>] 41.5	\[ \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 41.5	\[ \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \cdot \frac{\sqrt{2}}{B} \]
      unpow2 [=>] 41.5	\[ \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \cdot \frac{\sqrt{2}}{B} \]

   7. Applied egg-rr 37.0

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

   8. Simplified 37.0

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

      [Start] 37.0	\[ \sqrt{F \cdot \left(\left(A \cdot \left(A - A\right) - B \cdot B\right) \cdot \frac{1}{A + \mathsf{hypot}\left(A, B\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      associate-*r/ [=>] 37.0	\[ \sqrt{F \cdot \color{blue}{\frac{\left(A \cdot \left(A - A\right) - B \cdot B\right) \cdot 1}{A + \mathsf{hypot}\left(A, B\right)}}} \cdot \frac{\sqrt{2}}{B} \]
      *-rgt-identity [=>] 37.0	\[ \sqrt{F \cdot \frac{\color{blue}{A \cdot \left(A - A\right) - B \cdot B}}{A + \mathsf{hypot}\left(A, B\right)}} \cdot \frac{\sqrt{2}}{B} \]
      div-sub [=>] 37.0	\[ \sqrt{F \cdot \color{blue}{\left(\frac{A \cdot \left(A - A\right)}{A + \mathsf{hypot}\left(A, B\right)} - \frac{B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}\right)}} \cdot \frac{\sqrt{2}}{B} \]
      +-inverses [=>] 37.0	\[ \sqrt{F \cdot \left(\frac{A \cdot \color{blue}{0}}{A + \mathsf{hypot}\left(A, B\right)} - \frac{B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      mul0-rgt [=>] 37.0	\[ \sqrt{F \cdot \left(\frac{\color{blue}{0}}{A + \mathsf{hypot}\left(A, B\right)} - \frac{B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      +-inverses [<=] 37.0	\[ \sqrt{F \cdot \left(\frac{\color{blue}{A - A}}{A + \mathsf{hypot}\left(A, B\right)} - \frac{B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}\right)} \cdot \frac{\sqrt{2}}{B} \]
      div-sub [<=] 37.0	\[ \sqrt{F \cdot \color{blue}{\frac{\left(A - A\right) - B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}} \cdot \frac{\sqrt{2}}{B} \]
      +-inverses [=>] 37.0	\[ \sqrt{F \cdot \frac{\color{blue}{0} - B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}} \cdot \frac{\sqrt{2}}{B} \]
      neg-sub0 [<=] 37.0	\[ \sqrt{F \cdot \frac{\color{blue}{-B \cdot B}}{A + \mathsf{hypot}\left(A, B\right)}} \cdot \frac{\sqrt{2}}{B} \]
      distribute-rgt-neg-in [=>] 37.0	\[ \sqrt{F \cdot \frac{\color{blue}{B \cdot \left(-B\right)}}{A + \mathsf{hypot}\left(A, B\right)}} \cdot \frac{\sqrt{2}}{B} \]

   if -1.20000000000000003e-79 < B < -7.9999999999999998e-223

   1. Initial program 52.6

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

   2. Simplified 52.0

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

      [Start] 52.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 63.0

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

   4. Simplified 63.0

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

      [Start] 63.0	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 63.0	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 63.0	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 63.0	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      *-commutative [=>] 63.0	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}} \]
      unpow2 [=>] 63.0	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]
      unpow2 [=>] 63.0	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \]

   5. Taylor expanded in A around inf 59.8

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

   6. Simplified 59.8

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

      [Start] 59.8	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A}\right)} \]
      associate-*r/ [=>] 59.8	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}} \]
      unpow2 [=>] 59.8	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}} \]
      associate-*r* [=>] 59.8	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{\color{blue}{\left(-0.5 \cdot B\right) \cdot B}}{A}} \]

   7. Applied egg-rr 58.2

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

   if -7.9999999999999998e-223 < B < -1.75000000000000008e-265 or 9.50000000000000012e-264 < B < 1.14999999999999993e-120

   1. Initial program 53.2

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

   2. Simplified 46.8

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

      [Start] 53.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 48.9

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

   if -1.75000000000000008e-265 < B < 9.50000000000000012e-264

   1. Initial program 53.4

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

   2. Simplified 46.6

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

      [Start] 53.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 52.9

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Simplified 53.0

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof

      [Start] 52.9	\[ \frac{-\sqrt{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 53.0	\[ \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 53.0	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(F \cdot {C}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 53.0	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \color{blue}{\left(C \cdot C\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 49.7

      \[\leadsto \color{blue}{0 + \frac{\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot C}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Simplified 49.6

      \[\leadsto \color{blue}{C \cdot \frac{\sqrt{A \cdot \left(-16 \cdot F\right)}}{\mathsf{fma}\left(B, B, \left(A \cdot -4\right) \cdot C\right)}} \]
      Proof

      [Start] 49.7	\[ 0 + \frac{\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot C}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-lft-identity [=>] 49.7	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot C}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      associate-/l* [=>] 49.0	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{C}}} \]
      associate-/r/ [=>] 49.7	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot C} \]
      *-lft-identity [<=] 49.7	\[ \frac{\color{blue}{1 \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot C \]
      associate-*l/ [<=] 49.7	\[ \color{blue}{\left(\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}\right)} \cdot C \]
      *-commutative [=>] 49.7	\[ \color{blue}{C \cdot \left(\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}\right)} \]
      associate-*l/ [=>] 49.7	\[ C \cdot \color{blue}{\frac{1 \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-lft-identity [=>] 49.7	\[ C \cdot \frac{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 49.8	\[ C \cdot \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot F}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.8	\[ C \cdot \frac{\sqrt{\color{blue}{\left(A \cdot -16\right)} \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 49.6	\[ C \cdot \frac{\sqrt{\color{blue}{A \cdot \left(-16 \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 49.6	\[ C \cdot \frac{\sqrt{A \cdot \left(-16 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(-4 \cdot C\right)}\right)} \]
      associate-*r* [=>] 49.6	\[ C \cdot \frac{\sqrt{A \cdot \left(-16 \cdot F\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)} \]

   if 1.14999999999999993e-120 < B < 1450

   1. Initial program 44.3

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

   2. Simplified 44.0

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

      [Start] 44.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 50.1

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

   4. Simplified 50.1

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

      [Start] 50.1	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 50.1	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 50.1	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 50.1	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      *-commutative [=>] 50.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}} \]
      unpow2 [=>] 50.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]
      unpow2 [=>] 50.1	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \]

   5. Taylor expanded in A around inf 54.4

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

   6. Simplified 54.4

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

      [Start] 54.4	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A}\right)} \]
      associate-*r/ [=>] 54.4	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}} \]
      unpow2 [=>] 54.4	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}} \]
      associate-*r* [=>] 54.4	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{\color{blue}{\left(-0.5 \cdot B\right) \cdot B}}{A}} \]

   7. Applied egg-rr 52.1

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

   if 1450 < B

   1. Initial program 56.0

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

   2. Simplified 55.9

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

      [Start] 56.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 53.2

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

   4. Simplified 53.2

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

      [Start] 53.2	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 53.2	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      associate-*r/ [=>] 53.2	\[ \color{blue}{\frac{-1 \cdot \sqrt{2}}{B}} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      mul-1-neg [=>] 53.2	\[ \frac{\color{blue}{-\sqrt{2}}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \]
      *-commutative [=>] 53.2	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}} \]
      unpow2 [=>] 53.2	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]
      unpow2 [=>] 53.2	\[ \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \]

   5. Applied egg-rr 36.1

      \[\leadsto \frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 42.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.8 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-223}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-264}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-120}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot 2\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1450:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \frac{-\sqrt{F \cdot \left(-0.5 \cdot \left(B \cdot B\right)\right)}}{\sqrt{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	39.7
Cost	34252

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -4.9 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -2.75 \cdot 10^{-78}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(F \cdot t_0\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

Alternative 2
Error	39.7
Cost	27916

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -5.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -3.85 \cdot 10^{-78}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

Alternative 3
Error	39.1
Cost	27852

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-40}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

Alternative 4
Error	43.1
Cost	21084

\[\begin{array}{l} t_0 := C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -5.1 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-80}:\\ \;\;\;\;t_2 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-306}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{t_1} \cdot t_0\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{F \cdot \left(-0.5 \cdot \left(B \cdot B\right)\right)}}{\sqrt{A}}\\ \mathbf{elif}\;B \leq 840000000000:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \frac{1}{\frac{1}{A - \mathsf{hypot}\left(A, B\right)}}}\right)\\ \end{array} \]

Alternative 5
Error	43.1
Cost	20888

\[\begin{array}{l} t_0 := C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}\\ t_1 := -\sqrt{2}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ t_4 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -9 \cdot 10^{+140}:\\ \;\;\;\;t_3 \cdot t_4\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-80}:\\ \;\;\;\;t_4 \cdot \sqrt{F \cdot \frac{-B \cdot B}{A + \mathsf{hypot}\left(A, B\right)}}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-212}:\\ \;\;\;\;\frac{t_1 \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{t_2} \cdot t_0\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-19}:\\ \;\;\;\;t_4 \cdot \frac{-\sqrt{F \cdot \left(-0.5 \cdot \left(B \cdot B\right)\right)}}{\sqrt{A}}\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;\frac{t_0}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{t_1}{B}\\ \end{array} \]

Alternative 6
Error	43.5
Cost	20828

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_3 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -2.25 \cdot 10^{+43}:\\ \;\;\;\;t_1 \cdot t_3\\ \mathbf{elif}\;B \leq -5.2 \cdot 10^{-195}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-128}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-19}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;\frac{C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	43.3
Cost	20756

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_4 := C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-198}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-306}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_4\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;t_2 \cdot \frac{-\sqrt{F \cdot \left(-0.5 \cdot \left(B \cdot B\right)\right)}}{\sqrt{A}}\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;\frac{t_4}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Error	44.1
Cost	19972

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -2.25 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot t_2\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-128}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-19}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 9
Error	44.5
Cost	15368

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.5 \cdot 10^{+43}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-195}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-128}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-19}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1100000000000:\\ \;\;\;\;\frac{C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 10
Error	45.2
Cost	14684

\[\begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5.6 \cdot 10^{-80}:\\ \;\;\;\;t_1 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-216}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-128}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-18}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 11
Error	44.8
Cost	14552

\[\begin{array}{l} t_0 := C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-80}:\\ \;\;\;\;t_1 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-171}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{\left(B \cdot -0.5\right) \cdot \left(B \cdot F\right)}{A}}\right)\\ \mathbf{elif}\;B \leq -4.6 \cdot 10^{-178}:\\ \;\;\;\;t_1 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-20}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 12
Error	44.8
Cost	14552

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -2.8 \cdot 10^{-78}:\\ \;\;\;\;t_0 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq -6.4 \cdot 10^{-171}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{\frac{\left(B \cdot -0.5\right) \cdot \left(B \cdot F\right)}{A}}\right)\\ \mathbf{elif}\;B \leq -5.7 \cdot 10^{-179}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-128}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-20}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;\frac{C \cdot \sqrt{-16 \cdot \left(F \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 13
Error	46.2
Cost	14224

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{-80}:\\ \;\;\;\;t_0 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-223}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{\frac{\left(B \cdot -0.5\right) \cdot \left(B \cdot F\right)}{A}}\right)\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1760000000000:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 14
Error	46.2
Cost	14224

\[\begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.42 \cdot 10^{-80}:\\ \;\;\;\;t_1 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{t_0 \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}}{B}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+14}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 15
Error	46.0
Cost	14092

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-172}:\\ \;\;\;\;t_0 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 44000000000000:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{\frac{B \cdot -0.5}{\frac{\frac{A}{B}}{F}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 16
Error	45.9
Cost	13704

\[\begin{array}{l} \mathbf{if}\;B \leq -2.75 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 17
Error	45.8
Cost	13640

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -4.9 \cdot 10^{-172}:\\ \;\;\;\;t_0 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 18
Error	50.5
Cost	13380

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	55.4
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	57.0
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	59.5
Cost	8320

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0} \end{array} \]

Alternative 22
Error	61.8
Cost	8064

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

Alternative 23
Error	61.9
Cost	7808

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

Alternative 24
Error	62.6
Cost	7680

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

Alternative 25
Error	62.6
Cost	7680

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

Alternative 26
Error	63.0
Cost	7296

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

Reproduce

herbie shell --seed 2023018 
(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))))