ABCF->ab-angle b

Average Accuracy: 18.2% → 37.0%

Time: 41.8s

Precision: binary64

Cost: 27984

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{\left(C + \left(A - t_1\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -2.15 \cdot 10^{-123}:\\ \;\;\;\;\frac{t_2}{B + -4 \cdot \frac{C \cdot A}{B}}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{A \cdot -8} \cdot \left(-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{-69}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(C - t_1\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{-B}\\ \end{array} \]

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 (fma B B (* A (* C -4.0))))
        (t_1 (hypot B (- A C)))
        (t_2 (sqrt (* (+ C (- A t_1)) (* 2.0 F)))))
   (if (<= B -2.15e-123)
     (/ t_2 (+ B (* -4.0 (/ (* C A) B))))
     (if (<= B 2.7e-174)
       (/ (* (sqrt (* A -8.0)) (- (sqrt (* F (* C (+ A A)))))) t_0)
       (if (<= B 1.32e-69)
         (* C (/ (sqrt (* A (* F -16.0))) (fma B B (* C (* A -4.0)))))
         (if (<= B 2.5e+39)
           (/ (- (sqrt (* t_0 (* (* 2.0 F) (+ A (- C t_1)))))) t_0)
           (/ t_2 (- 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 = hypot(B, (A - C));
	double t_2 = sqrt(((C + (A - t_1)) * (2.0 * F)));
	double tmp;
	if (B <= -2.15e-123) {
		tmp = t_2 / (B + (-4.0 * ((C * A) / B)));
	} else if (B <= 2.7e-174) {
		tmp = (sqrt((A * -8.0)) * -sqrt((F * (C * (A + A))))) / t_0;
	} else if (B <= 1.32e-69) {
		tmp = C * (sqrt((A * (F * -16.0))) / fma(B, B, (C * (A * -4.0))));
	} else if (B <= 2.5e+39) {
		tmp = -sqrt((t_0 * ((2.0 * F) * (A + (C - t_1))))) / t_0;
	} else {
		tmp = t_2 / -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 = hypot(B, Float64(A - C))
	t_2 = sqrt(Float64(Float64(C + Float64(A - t_1)) * Float64(2.0 * F)))
	tmp = 0.0
	if (B <= -2.15e-123)
		tmp = Float64(t_2 / Float64(B + Float64(-4.0 * Float64(Float64(C * A) / B))));
	elseif (B <= 2.7e-174)
		tmp = Float64(Float64(sqrt(Float64(A * -8.0)) * Float64(-sqrt(Float64(F * Float64(C * Float64(A + A)))))) / t_0);
	elseif (B <= 1.32e-69)
		tmp = Float64(C * Float64(sqrt(Float64(A * Float64(F * -16.0))) / fma(B, B, Float64(C * Float64(A * -4.0)))));
	elseif (B <= 2.5e+39)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(Float64(2.0 * F) * Float64(A + Float64(C - t_1)))))) / t_0);
	else
		tmp = Float64(t_2 / Float64(-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[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(C + N[(A - t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -2.15e-123], N[(t$95$2 / N[(B + N[(-4.0 * N[(N[(C * A), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.7e-174], N[(N[(N[Sqrt[N[(A * -8.0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1.32e-69], 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, 2.5e+39], N[((-N[Sqrt[N[(t$95$0 * N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[(C - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(t$95$2 / (-B)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -2.15000000000000016e-123

   1. Initial program 18.6%

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

   2. Simplified 23.3%

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

      [Start] 18.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 1.3%

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

      [Start] 23.3

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

      add-sqr-sqrt [=>] 0.6

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

   4. Simplified 1.3%

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

      [Start] 1.3	\[ \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r/ [=>] 1.1	\[ \color{blue}{\frac{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      /-rgt-identity [=>] 1.1	\[ \frac{\color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [<=] 1.1	\[ \frac{\color{blue}{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 1.3	\[ \color{blue}{\frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}} \]
      +-commutative [=>] 1.3	\[ \frac{\sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate--l+ [=>] 1.3	\[ \frac{\sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate-*r* [=>] 1.3	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate-*r* [=>] 1.3	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{\left(A \cdot C\right) \cdot -4}}\right)}} \]

   5. Taylor expanded in A around 0 41.3%

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

   if -2.15000000000000016e-123 < B < 2.69999999999999988e-174

   1. Initial program 17.5%

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

   2. Simplified 27.4%

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

      [Start] 17.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 20.3%

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

   4. Simplified 20.4%

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

      [Start] 20.3	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 20.4	\[ \frac{-\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 20.4	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 20.4	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      mul-1-neg [=>] 20.4	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 22.1%

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

      [Start] 20.4	\[ \frac{-\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-prod [=>] 19.1	\[ \frac{-\color{blue}{\sqrt{-8 \cdot A} \cdot \sqrt{\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 19.1	\[ \frac{-\color{blue}{\sqrt{\left(F \cdot C\right) \cdot \left(A - \left(-A\right)\right)} \cdot \sqrt{-8 \cdot A}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 22.1	\[ \frac{-\sqrt{\color{blue}{F \cdot \left(C \cdot \left(A - \left(-A\right)\right)\right)}} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      neg-mul-1 [=>] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \left(A - \color{blue}{-1 \cdot A}\right)\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      cancel-sign-sub-inv [=>] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \color{blue}{\left(A + \left(--1\right) \cdot A\right)}\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      metadata-eval [=>] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \left(A + \color{blue}{1} \cdot A\right)\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-un-lft-identity [<=] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \left(A + \color{blue}{A}\right)\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Simplified 22.1%

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

      [Start] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)} \cdot \sqrt{-8 \cdot A}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 22.1	\[ \frac{-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)} \cdot \sqrt{\color{blue}{A \cdot -8}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 2.69999999999999988e-174 < B < 1.32000000000000007e-69

   1. Initial program 21.5%

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

   2. Simplified 29.1%

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

      [Start] 21.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 14.6%

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

   4. Simplified 14.6%

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

      [Start] 14.6	\[ \frac{-\sqrt{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 14.6	\[ \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 14.6	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(F \cdot {C}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      unpow2 [=>] 14.6	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \color{blue}{\left(C \cdot C\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Applied egg-rr 23.2%

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

      [Start] 14.6	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      distribute-frac-neg [=>] 14.6	\[ \color{blue}{-\frac{\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      neg-sub0 [=>] 14.6	\[ \color{blue}{0 - \frac{\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      add-sqr-sqrt [=>] 14.6	\[ 0 - \frac{\sqrt{\color{blue}{\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)} \cdot \sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqr-neg [<=] 14.6	\[ 0 - \frac{\sqrt{\color{blue}{\left(-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\right) \cdot \left(-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-unprod [<=] 1.0	\[ 0 - \frac{\color{blue}{\sqrt{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}} \cdot \sqrt{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      add-sqr-sqrt [<=] 1.4	\[ 0 - \frac{\color{blue}{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      distribute-frac-neg [=>] 1.4	\[ 0 - \color{blue}{\left(-\frac{\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      neg-sub0 [=>] 1.4	\[ 0 - \color{blue}{\left(0 - \frac{\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]

   6. Simplified 23.5%

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

      [Start] 23.2	\[ 0 + \frac{\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot C}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      +-lft-identity [=>] 23.2	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot C}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      associate-/l* [=>] 24.6	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{C}}} \]
      associate-/r/ [=>] 23.5	\[ \color{blue}{\frac{\sqrt{-16 \cdot \left(A \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot C} \]
      *-lft-identity [<=] 23.5	\[ \frac{\color{blue}{1 \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot C \]
      associate-*l/ [<=] 23.5	\[ \color{blue}{\left(\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}\right)} \cdot C \]
      *-commutative [=>] 23.5	\[ \color{blue}{C \cdot \left(\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}\right)} \]
      associate-*l/ [=>] 23.5	\[ C \cdot \color{blue}{\frac{1 \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-lft-identity [=>] 23.5	\[ C \cdot \frac{\color{blue}{\sqrt{-16 \cdot \left(A \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 23.5	\[ C \cdot \frac{\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot F}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 23.5	\[ C \cdot \frac{\sqrt{\color{blue}{\left(A \cdot -16\right)} \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*l* [=>] 23.5	\[ C \cdot \frac{\sqrt{\color{blue}{A \cdot \left(-16 \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [=>] 23.5	\[ C \cdot \frac{\sqrt{A \cdot \left(-16 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(-4 \cdot C\right)}\right)} \]
      associate-*r* [=>] 23.5	\[ C \cdot \frac{\sqrt{A \cdot \left(-16 \cdot F\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)} \]

   if 1.32000000000000007e-69 < B < 2.50000000000000008e39

   1. Initial program 35.1%

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

   2. Simplified 42.5%

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

      [Start] 35.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 2.50000000000000008e39 < B

   1. Initial program 10.5%

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

   2. Simplified 12.5%

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

      [Start] 10.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 1.5%

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

      [Start] 12.5

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

      add-sqr-sqrt [=>] 0.2

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

   4. Simplified 1.4%

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

      [Start] 1.5	\[ \frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r/ [=>] 1.1	\[ \color{blue}{\frac{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{1} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      /-rgt-identity [=>] 1.1	\[ \frac{\color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      *-commutative [<=] 1.1	\[ \frac{\color{blue}{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)} \cdot \mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-/l* [=>] 1.4	\[ \color{blue}{\frac{\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}}} \]
      +-commutative [=>] 1.4	\[ \frac{\sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate--l+ [=>] 1.4	\[ \frac{\sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate-*r* [=>] 1.4	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}} \]
      associate-*r* [=>] 1.4	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\mathsf{hypot}\left(B, \sqrt{\color{blue}{\left(A \cdot C\right) \cdot -4}}\right)}} \]

   5. Taylor expanded in B around -inf 51.7%

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

   6. Simplified 51.7%

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

      [Start] 51.7	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{-1 \cdot B} \]
      mul-1-neg [=>] 51.7	\[ \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{-B}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.15 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{B + -4 \cdot \frac{C \cdot A}{B}}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{A \cdot -8} \cdot \left(-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{-69}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	37.0%
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \sqrt{t_1 \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_2}{B + -4 \cdot \frac{C \cdot A}{B}}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-183}:\\ \;\;\;\;\frac{\sqrt{A \cdot -8} \cdot \left(-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-69}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(t_1 \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{-B}\\ \end{array} \]

Alternative 2
Accuracy	35.1%
Cost	21008

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \frac{\sqrt{A \cdot -8} \cdot \left(-\sqrt{F \cdot \left(C \cdot \left(A + A\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{t_0}{B + -4 \cdot \frac{C \cdot A}{B}}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-60}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 13500000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 3
Accuracy	24.5%
Cost	14356

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{if}\;A \leq -8 \cdot 10^{+32}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.42 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 2.3 \cdot 10^{+61}:\\ \;\;\;\;C \cdot \frac{\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 4
Accuracy	35.4%
Cost	14340

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{-139}:\\ \;\;\;\;\frac{t_0}{B + -4 \cdot \frac{C \cdot A}{B}}\\ \mathbf{elif}\;B \leq 420000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 5
Accuracy	35.7%
Cost	14024

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;B \leq 4500000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 6
Accuracy	26.1%
Cost	13828

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A - \left(B - C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 7
Accuracy	22.5%
Cost	13580

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ \mathbf{if}\;A \leq -1.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -7.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 3.35 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 8
Accuracy	19.2%
Cost	8716

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -3.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 2.3 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	19.8%
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := \left(C \cdot A\right) \cdot 4\\ t_2 := \sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}\\ t_3 := B \cdot B + t_0\\ \mathbf{if}\;C \leq -1.25 \cdot 10^{-261}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_2}{t_1 - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	17.7%
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ \mathbf{if}\;A \leq 4.8 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(B + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 11
Accuracy	18.3%
Cost	7812

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq 1.05 \cdot 10^{-229}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 12
Accuracy	18.3%
Cost	7684

\[\begin{array}{l} \mathbf{if}\;A \leq 8 \cdot 10^{-233}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{\left(C \cdot A\right) \cdot 4}\\ \end{array} \]

Alternative 13
Accuracy	13.9%
Cost	7360

\[\frac{\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{\left(C \cdot A\right) \cdot 4} \]

Alternative 14
Accuracy	2.5%
Cost	6848

\[-0.5 \cdot \sqrt{2 \cdot \frac{F}{C}} \]

Alternative 15
Accuracy	1.2%
Cost	6720

\[\sqrt{\frac{F \cdot 0.5}{C}} \]

Reproduce

herbie shell --seed 2023135 
(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))))