ABCF->ab-angle a

Average Error: 52.4 → 45.8

Time: 37.8s

Precision: binary64

Cost: 203088

Math

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

↓

\[\begin{array}{l} t_0 := {\left(A - C\right)}^{2}\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{t_0 + {B}^{2}}\right)}}{t_1}\\ t_3 := 4 \cdot \left(A \cdot C\right)\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t_3\right) \cdot F\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + t_0}\right)\right)}}{t_3 - {B}^{2}}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+251}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot B}\right)\\ \end{array} \]

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 (pow (- A C) 2.0))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt (* (* 2.0 (* t_1 F)) (+ (+ A C) (sqrt (+ t_0 (pow B 2.0)))))))
          t_1))
        (t_3 (* 4.0 (* A C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_3) F))
            (+ A (+ C (sqrt (+ (pow B 2.0) t_0))))))
          (- t_3 (pow B 2.0)))))
   (if (<= t_2 (- INFINITY))
     (* -1.0 (* (sqrt 2.0) (sqrt (/ F B))))
     (if (<= t_2 -1e-220)
       t_4
       (if (<= t_2 5e-151)
         (/ (sqrt (* 2.0 (* (pow A 2.0) (* C (* -8.0 F))))) (* C (* A 4.0)))
         (if (<= t_2 5e+251)
           t_4
           (* (/ (sqrt 2.0) B) (* -1.0 (sqrt (* F 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 = pow((A - C), 2.0);
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((t_0 + pow(B, 2.0)))))) / t_1;
	double t_3 = 4.0 * (A * C);
	double t_4 = sqrt(((2.0 * ((pow(B, 2.0) - t_3) * F)) * (A + (C + sqrt((pow(B, 2.0) + t_0)))))) / (t_3 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -1.0 * (sqrt(2.0) * sqrt((F / B)));
	} else if (t_2 <= -1e-220) {
		tmp = t_4;
	} else if (t_2 <= 5e-151) {
		tmp = sqrt((2.0 * (pow(A, 2.0) * (C * (-8.0 * F))))) / (C * (A * 4.0));
	} else if (t_2 <= 5e+251) {
		tmp = t_4;
	} else {
		tmp = (sqrt(2.0) / B) * (-1.0 * sqrt((F * B)));
	}
	return tmp;
}

Java

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

↓

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow((A - C), 2.0);
	double t_1 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_2 = -Math.sqrt(((2.0 * (t_1 * F)) * ((A + C) + Math.sqrt((t_0 + Math.pow(B, 2.0)))))) / t_1;
	double t_3 = 4.0 * (A * C);
	double t_4 = Math.sqrt(((2.0 * ((Math.pow(B, 2.0) - t_3) * F)) * (A + (C + Math.sqrt((Math.pow(B, 2.0) + t_0)))))) / (t_3 - Math.pow(B, 2.0));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 * (Math.sqrt(2.0) * Math.sqrt((F / B)));
	} else if (t_2 <= -1e-220) {
		tmp = t_4;
	} else if (t_2 <= 5e-151) {
		tmp = Math.sqrt((2.0 * (Math.pow(A, 2.0) * (C * (-8.0 * F))))) / (C * (A * 4.0));
	} else if (t_2 <= 5e+251) {
		tmp = t_4;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (-1.0 * Math.sqrt((F * B)));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	return -math.sqrt(((2.0 * ((math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / (math.pow(B, 2.0) - ((4.0 * A) * C))

↓

def code(A, B, C, F):
	t_0 = math.pow((A - C), 2.0)
	t_1 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_2 = -math.sqrt(((2.0 * (t_1 * F)) * ((A + C) + math.sqrt((t_0 + math.pow(B, 2.0)))))) / t_1
	t_3 = 4.0 * (A * C)
	t_4 = math.sqrt(((2.0 * ((math.pow(B, 2.0) - t_3) * F)) * (A + (C + math.sqrt((math.pow(B, 2.0) + t_0)))))) / (t_3 - math.pow(B, 2.0))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -1.0 * (math.sqrt(2.0) * math.sqrt((F / B)))
	elif t_2 <= -1e-220:
		tmp = t_4
	elif t_2 <= 5e-151:
		tmp = math.sqrt((2.0 * (math.pow(A, 2.0) * (C * (-8.0 * F))))) / (C * (A * 4.0))
	elif t_2 <= 5e+251:
		tmp = t_4
	else:
		tmp = (math.sqrt(2.0) / B) * (-1.0 * math.sqrt((F * 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 = Float64(A - C) ^ 2.0
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64(t_0 + (B ^ 2.0))))))) / t_1)
	t_3 = Float64(4.0 * Float64(A * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_3) * F)) * Float64(A + Float64(C + sqrt(Float64((B ^ 2.0) + t_0)))))) / Float64(t_3 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-1.0 * Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	elseif (t_2 <= -1e-220)
		tmp = t_4;
	elseif (t_2 <= 5e-151)
		tmp = Float64(sqrt(Float64(2.0 * Float64((A ^ 2.0) * Float64(C * Float64(-8.0 * F))))) / Float64(C * Float64(A * 4.0)));
	elseif (t_2 <= 5e+251)
		tmp = t_4;
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-1.0 * sqrt(Float64(F * B))));
	end
	return tmp
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(((2.0 * (((B ^ 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / ((B ^ 2.0) - ((4.0 * A) * C));
end

↓

function tmp_2 = code(A, B, C, F)
	t_0 = (A - C) ^ 2.0;
	t_1 = (B ^ 2.0) - ((4.0 * A) * C);
	t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((t_0 + (B ^ 2.0)))))) / t_1;
	t_3 = 4.0 * (A * C);
	t_4 = sqrt(((2.0 * (((B ^ 2.0) - t_3) * F)) * (A + (C + sqrt(((B ^ 2.0) + t_0)))))) / (t_3 - (B ^ 2.0));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -1.0 * (sqrt(2.0) * sqrt((F / B)));
	elseif (t_2 <= -1e-220)
		tmp = t_4;
	elseif (t_2 <= 5e-151)
		tmp = sqrt((2.0 * ((A ^ 2.0) * (C * (-8.0 * F))))) / (C * (A * 4.0));
	elseif (t_2 <= 5e+251)
		tmp = t_4;
	else
		tmp = (sqrt(2.0) / B) * (-1.0 * sqrt((F * B)));
	end
	tmp_2 = 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[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(t$95$0 + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-1.0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-220], t$95$4, If[LessEqual[t$95$2, 5e-151], N[(N[Sqrt[N[(2.0 * N[(N[Power[A, 2.0], $MachinePrecision] * N[(C * N[(-8.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+251], t$95$4, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(-1.0 * N[Sqrt[N[(F * B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 64.0	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]
      rational.json-simplify-40 [<=] 64.0	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around 0 63.0

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

   4. Simplified 63.0

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

      [Start] 63.0	\[ \frac{\sqrt{2 \cdot \left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{\sqrt{2 \cdot \left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot \color{blue}{\left({B}^{2} \cdot F\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-31 [=>] 63.0	\[ \frac{\sqrt{2 \cdot \color{blue}{\left({B}^{2} \cdot \left(\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{\sqrt{2 \cdot \left({B}^{2} \cdot \color{blue}{\left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]

   5. Taylor expanded in C around 0 56.5

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -9.99999999999999992e-221 or 5.00000000000000003e-151 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 5.0000000000000005e251

   1. Initial program 1.7

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

   2. Simplified 1.7

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

      [Start] 1.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 1.7	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 1.7	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   if -9.99999999999999992e-221 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 5.00000000000000003e-151

   1. Initial program 60.8

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

   2. Simplified 58.6

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

      [Start] 60.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 60.8	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 60.8	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]
      rational.json-simplify-40 [<=] 60.8	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 53.6

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

   4. Simplified 53.6

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

      [Start] 53.6	\[ \frac{\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-31 [=>] 53.6	\[ \frac{\sqrt{2 \cdot \color{blue}{\left({A}^{2} \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-31 [=>] 53.6	\[ \frac{\sqrt{2 \cdot \left({A}^{2} \cdot \color{blue}{\left(C \cdot \left(-8 \cdot F\right)\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]

   5. Taylor expanded in A around inf 53.4

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

   6. Simplified 53.4

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

      [Start] 53.4	\[ \frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-31 [=>] 53.4	\[ \frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{\color{blue}{A \cdot \left(4 \cdot C\right)}} \]
      rational.json-simplify-2 [<=] 53.4	\[ \frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{A \cdot \color{blue}{\left(C \cdot 4\right)}} \]
      rational.json-simplify-31 [=>] 53.4	\[ \frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{\color{blue}{C \cdot \left(A \cdot 4\right)}} \]

   if 5.0000000000000005e251 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 63.9

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

   2. Simplified 63.9

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

      [Start] 63.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 63.9	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 63.9	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   3. Taylor expanded in A around 0 63.6

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

   4. Simplified 63.6

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

      [Start] 63.6	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      rational.json-simplify-31 [=>] 63.6	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 63.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

   5. Taylor expanded in C around 0 55.6

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \color{blue}{\sqrt{F \cdot B}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot B}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	48.3
Cost	47108

\[\begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(C + \left(A + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot \sqrt{\frac{F}{B}}\right) + \sqrt{F \cdot B}\right)\right)\\ \end{array} \]

Alternative 2
Error	48.5
Cost	40908

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} + \left(A \cdot C\right) \cdot -4\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-8 \cdot \left({C}^{2} \cdot \left(F \cdot A\right)\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+144}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + \sqrt{F \cdot B}\right)\right)\\ \end{array} \]

Alternative 3
Error	49.8
Cost	40004

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \left(4 \cdot A\right) \cdot C - {B}^{2}\\ t_2 := \sqrt{\frac{F}{B}}\\ \mathbf{if}\;B \leq -1.66 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right) + C \cdot \left({B}^{2} \cdot \left(4 \cdot F\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \left(-1 \cdot t_0\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_2\right) + \sqrt{F \cdot B}\right)\right)\\ \end{array} \]

Alternative 4
Error	50.0
Cost	33356

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{\frac{F}{B}}\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right) \cdot \left(A + \left(C + B \cdot -1\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right) + C \cdot \left({B}^{2} \cdot \left(4 \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 10^{+142}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \left(-1 \cdot t_1\right)\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_2\right) + \sqrt{F \cdot B}\right)\right)\\ \end{array} \]

Alternative 5
Error	50.2
Cost	27656

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot B}\\ t_3 := \sqrt{\frac{F}{B}}\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right) \cdot \left(A + \left(C + B \cdot -1\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right) + C \cdot \left({B}^{2} \cdot \left(4 \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+218}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_3\right) + t_2\right)\right)\\ \end{array} \]

Alternative 6
Error	50.2
Cost	21188

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := 4 \cdot \left(A \cdot C\right)\\ t_3 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t_2\right) \cdot F\right)\right) \cdot \left(A + \left(C + B \cdot -1\right)\right)}}{t_2 - {B}^{2}}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_3\right)\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_3\right)\right)\\ \end{array} \]

Alternative 7
Error	51.0
Cost	20816

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{3} \cdot \left(-1 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_2\right)\right)\\ \end{array} \]

Alternative 8
Error	50.9
Cost	20816

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{3} \cdot \left(-1 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left({A}^{2} \cdot \left(-16 \cdot F\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+218}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_2\right)\right)\\ \end{array} \]

Alternative 9
Error	50.9
Cost	20816

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{3} \cdot \left(-1 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-8 \cdot \left({C}^{2} \cdot \left(F \cdot A\right)\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+218}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_2\right)\right)\\ \end{array} \]

Alternative 10
Error	50.7
Cost	20816

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{F \cdot B}\\ t_2 := \left(4 \cdot A\right) \cdot C - {B}^{2}\\ t_3 := \sqrt{\frac{F}{B}}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{3} \cdot \left(-1 \cdot F\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 7.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+194}:\\ \;\;\;\;t_0 \cdot \left(-1 \cdot t_1\right)\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_3\right) + t_1\right)\right)\\ \end{array} \]

Alternative 11
Error	50.4
Cost	20816

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \left(4 \cdot A\right) \cdot C - {B}^{2}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(C + B \cdot -1\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({C}^{2} \cdot F\right) \cdot \left(-8 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+194}:\\ \;\;\;\;t_2 \cdot \left(-1 \cdot t_3\right)\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+218}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_3\right)\right)\\ \end{array} \]

Alternative 12
Error	52.1
Cost	20684

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}}\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot B}\\ \mathbf{if}\;B \leq 1.85 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+194}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot t_2\right)\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-1 \cdot \left(0.5 \cdot \left(C \cdot t_0\right) + t_2\right)\right)\\ \end{array} \]

Alternative 13
Error	52.1
Cost	13956

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq 1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({A}^{2} \cdot \left(C \cdot \left(-8 \cdot F\right)\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+194}:\\ \;\;\;\;t_0 \cdot \left(-1 \cdot \sqrt{F \cdot B}\right)\\ \mathbf{elif}\;B \leq 6.7 \cdot 10^{+217}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-1 \cdot \sqrt{F \cdot \left(C + B\right)}\right)\\ \end{array} \]

Alternative 14
Error	53.1
Cost	13508

\[\begin{array}{l} \mathbf{if}\;F \leq 20000000000:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 15
Error	55.5
Cost	13248

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

Reproduce

herbie shell --seed 2023053 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))