ABCF->ab-angle b

Average Error: 52.7 → 40.6

Time: 51.2s

Precision: binary64

Cost: 27720

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -4.4 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(\left(t_1 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(t_1 - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

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 A (* C -4.0) (* B B))) (t_1 (hypot B (- A C))))
   (if (<= B -4.4e+61)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B 4.8e-199)
       (/ (- (sqrt (* t_0 (* (- (- t_1 C) A) (* F -2.0))))) t_0)
       (if (<= B 1.4e-148)
         (/ (- (sqrt (* -8.0 (* (* A (+ C C)) (* C F))))) t_0)
         (if (<= B 5.5e-15)
           (/ (- (sqrt (* (* A -8.0) (* (* C F) (+ A A))))) t_0)
           (/
            (- (sqrt (* (- t_1 (+ A C)) (* F -2.0))))
            (fma -2.0 (/ A (/ B C)) B))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B * B));
	double t_1 = hypot(B, (A - C));
	double tmp;
	if (B <= -4.4e+61) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= 4.8e-199) {
		tmp = -sqrt((t_0 * (((t_1 - C) - A) * (F * -2.0)))) / t_0;
	} else if (B <= 1.4e-148) {
		tmp = -sqrt((-8.0 * ((A * (C + C)) * (C * F)))) / t_0;
	} else if (B <= 5.5e-15) {
		tmp = -sqrt(((A * -8.0) * ((C * F) * (A + A)))) / t_0;
	} else {
		tmp = -sqrt(((t_1 - (A + C)) * (F * -2.0))) / fma(-2.0, (A / (B / C)), B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_1 = hypot(B, Float64(A - C))
	tmp = 0.0
	if (B <= -4.4e+61)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= 4.8e-199)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(Float64(t_1 - C) - A) * Float64(F * -2.0))))) / t_0);
	elseif (B <= 1.4e-148)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(A * Float64(C + C)) * Float64(C * F))))) / t_0);
	elseif (B <= 5.5e-15)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(Float64(C * F) * Float64(A + A))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_1 - Float64(A + C)) * Float64(F * -2.0)))) / fma(-2.0, Float64(A / Float64(B / C)), B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B, -4.4e+61], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 4.8e-199], N[((-N[Sqrt[N[(t$95$0 * N[(N[(N[(t$95$1 - C), $MachinePrecision] - A), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.4e-148], N[((-N[Sqrt[N[(-8.0 * N[(N[(A * N[(C + C), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 5.5e-15], N[((-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(C * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(t$95$1 - N[(A + C), $MachinePrecision]), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * N[(A / N[(B / C), $MachinePrecision]), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -4.4000000000000001e61

   1. Initial program 58.0

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

   2. Simplified 57.0

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

      [Start] 58.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around -inf 58.8

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

   4. Taylor expanded in C around 0 63.1

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

   5. Simplified 63.1

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

      [Start] 63.1	\[ \frac{-\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*l* [=>] 63.1	\[ \frac{-\color{blue}{\sqrt{2} \cdot \left(B \cdot \sqrt{\left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right) \cdot F}\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      *-commutative [=>] 63.1	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{\color{blue}{F \cdot \left(A + \left(0.5 \cdot \frac{{A}^{2}}{B} + B\right)\right)}}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      fma-def [=>] 63.1	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \color{blue}{\mathsf{fma}\left(0.5, \frac{{A}^{2}}{B}, B\right)}\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 63.1	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \mathsf{fma}\left(0.5, \frac{\color{blue}{A \cdot A}}{B}, B\right)\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   6. Taylor expanded in A around 0 33.9

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

   7. Simplified 33.9

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

      [Start] 33.9	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      associate-*r* [=>] 33.9	\[ \color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
      neg-mul-1 [<=] 33.9	\[ \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]
      *-commutative [=>] 33.9	\[ \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)} \]

   if -4.4000000000000001e61 < B < 4.79999999999999991e-199

   1. Initial program 49.8

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

   2. Simplified 44.8

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

      [Start] 49.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 4.79999999999999991e-199 < B < 1.4e-148

   1. Initial program 51.4

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

   2. Simplified 46.6

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

      [Start] 51.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 50.9

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

   4. Simplified 48.8

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

      [Start] 50.9	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*r* [=>] 48.8	\[ \frac{-\sqrt{-8 \cdot \color{blue}{\left(\left(A \cdot \left(C - -1 \cdot C\right)\right) \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 48.8	\[ \frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 1.4e-148 < B < 5.5000000000000002e-15

   1. Initial program 46.8

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

   2. Simplified 41.3

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

      [Start] 46.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 53.0

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

   4. Simplified 53.0

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

      [Start] 53.0	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*r* [=>] 53.0	\[ \frac{-\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      *-commutative [=>] 53.0	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 53.0	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 5.5000000000000002e-15 < B

   1. Initial program 55.6

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

   2. Simplified 53.1

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

      [Start] 55.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 48.8

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

   4. Simplified 48.7

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

      [Start] 48.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 48.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 48.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 48.8

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

   6. Simplified 47.6

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

      [Start] 48.8	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 48.8	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 47.6	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 47.6	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 47.6	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 47.6	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around 0 34.2

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

   8. Simplified 33.3

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

      [Start] 34.2	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{-2 \cdot \frac{A \cdot C}{B} + B} \]
      fma-def [=>] 34.2	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{\mathsf{fma}\left(-2, \frac{A \cdot C}{B}, B\right)}} \]
      associate-/l* [=>] 33.3	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-2, \color{blue}{\frac{A}{\frac{B}{C}}}, B\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 40.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.4 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(\mathsf{hypot}\left(B, A - C\right) - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	39.9
Cost	155788

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{\left(-2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)}\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - t_0\right)\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_1}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t_3 \cdot \left(\left(A + C\right) - t_0\right)} \cdot \left(-\sqrt{2 \cdot F}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 2
Error	40.2
Cost	28172

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -8.6 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(\left(t_2 - C\right) - A\right) \cdot \left(F \cdot -2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C - \mathsf{fma}\left(-1, C, 0.5 \cdot \frac{B \cdot B}{A - C}\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(t_2 - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 3
Error	39.4
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 4
Error	40.1
Cost	27208

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{t_0}{\sqrt{t_2}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-61}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 5
Error	40.8
Cost	21704

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -9 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(A + \left(C - t_1\right)\right) \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right) + F \cdot \left(B \cdot B\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(t_1 - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 6
Error	41.9
Cost	21072

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_0}{2 \cdot \frac{A \cdot C}{B} - B}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 7
Error	41.9
Cost	21072

\[\begin{array}{l} t_0 := \frac{A}{\frac{B}{C}}\\ t_1 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+209}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-1, B, 2 \cdot t_0\right)}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-195}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-2, t_0, B\right)}\\ \end{array} \]

Alternative 8
Error	41.6
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 9
Error	41.5
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{\mathsf{fma}\left(-2, \frac{A}{\frac{B}{C}}, B\right)}\\ \end{array} \]

Alternative 10
Error	42.7
Cost	14608

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -7 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-192}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(C \cdot \left(\left(C + C\right) \cdot \left(A \cdot F\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 11
Error	42.4
Cost	14608

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -4 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-199}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 12
Error	42.0
Cost	14608

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-194}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 13
Error	42.0
Cost	14608

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_0}{2 \cdot \frac{A \cdot C}{B} - B}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 14
Error	41.7
Cost	14156

\[\begin{array}{l} t_0 := -\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+208}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -3.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{t_0}{-B}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B}\\ \end{array} \]

Alternative 15
Error	42.7
Cost	14024

\[\begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) - \left(A + C\right)\right) \cdot \left(F \cdot -2\right)}}{B}\\ \end{array} \]

Alternative 16
Error	49.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 17
Error	52.7
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -4.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -5 \cdot 10^{-249}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 18
Error	53.1
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.55 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.12 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 19
Error	55.8
Cost	8452

\[\begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B - C\right) - A\right) \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 20
Error	57.0
Cost	7552

\[\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C + C\right) \cdot \left(C \cdot F\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)} \]

Alternative 21
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023016 
(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))))