ABCF->ab-angle a

Average Error: 52.1 → 41.4

Time: 1.1min

Precision: binary64

Cost: 217172

Math

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

↓

\[\begin{array}{l} t_0 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_1 := {\left(A - C\right)}^{2}\\ t_2 := A \cdot \left(4 \cdot C\right)\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{t_1 + {B}^{2}}\right)}}{t_3}\\ t_5 := A + \left(C + \sqrt{{B}^{2} + t_1}\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_2\right) \cdot F\right) \cdot t_5\right)}}{t_2 - {B}^{2}}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(t_5 \cdot \left(2 \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

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 (- (pow B 2.0) (* 4.0 (* A C))))
        (t_1 (pow (- A C) 2.0))
        (t_2 (* A (* 4.0 C)))
        (t_3 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt (* (* 2.0 (* t_3 F)) (+ (+ A C) (sqrt (+ t_1 (pow B 2.0)))))))
          t_3))
        (t_5 (+ A (+ C (sqrt (+ (pow B 2.0) t_1))))))
   (if (<= t_4 (- INFINITY))
     (- (sqrt (- (/ F A))))
     (if (<= t_4 -2e-202)
       (/ (sqrt (* 2.0 (* (* (- (pow B 2.0) t_2) F) t_5))) (- t_2 (pow B 2.0)))
       (if (<= t_4 0.0)
         (- (sqrt (- (/ F C))))
         (if (<= t_4 2e+277)
           (/ (- (sqrt (* t_0 (* t_5 (* 2.0 F))))) t_0)
           (if (<= t_4 INFINITY)
             (/
              1.0
              (/
               (- (* A (* C 4.0)) (pow B 2.0))
               (* C (sqrt (* (* F A) -16.0)))))
             (- (sqrt (* 2.0 (/ 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(B, 2.0) - (4.0 * (A * C));
	double t_1 = pow((A - C), 2.0);
	double t_2 = A * (4.0 * C);
	double t_3 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((t_1 + pow(B, 2.0)))))) / t_3;
	double t_5 = A + (C + sqrt((pow(B, 2.0) + t_1)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -sqrt(-(F / A));
	} else if (t_4 <= -2e-202) {
		tmp = sqrt((2.0 * (((pow(B, 2.0) - t_2) * F) * t_5))) / (t_2 - pow(B, 2.0));
	} else if (t_4 <= 0.0) {
		tmp = -sqrt(-(F / C));
	} else if (t_4 <= 2e+277) {
		tmp = -sqrt((t_0 * (t_5 * (2.0 * F)))) / t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = 1.0 / (((A * (C * 4.0)) - pow(B, 2.0)) / (C * sqrt(((F * A) * -16.0))));
	} else {
		tmp = -sqrt((2.0 * (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(B, 2.0) - (4.0 * (A * C));
	double t_1 = Math.pow((A - C), 2.0);
	double t_2 = A * (4.0 * C);
	double t_3 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -Math.sqrt(((2.0 * (t_3 * F)) * ((A + C) + Math.sqrt((t_1 + Math.pow(B, 2.0)))))) / t_3;
	double t_5 = A + (C + Math.sqrt((Math.pow(B, 2.0) + t_1)));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt(-(F / A));
	} else if (t_4 <= -2e-202) {
		tmp = Math.sqrt((2.0 * (((Math.pow(B, 2.0) - t_2) * F) * t_5))) / (t_2 - Math.pow(B, 2.0));
	} else if (t_4 <= 0.0) {
		tmp = -Math.sqrt(-(F / C));
	} else if (t_4 <= 2e+277) {
		tmp = -Math.sqrt((t_0 * (t_5 * (2.0 * F)))) / t_0;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (((A * (C * 4.0)) - Math.pow(B, 2.0)) / (C * Math.sqrt(((F * A) * -16.0))));
	} else {
		tmp = -Math.sqrt((2.0 * (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(B, 2.0) - (4.0 * (A * C))
	t_1 = math.pow((A - C), 2.0)
	t_2 = A * (4.0 * C)
	t_3 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_4 = -math.sqrt(((2.0 * (t_3 * F)) * ((A + C) + math.sqrt((t_1 + math.pow(B, 2.0)))))) / t_3
	t_5 = A + (C + math.sqrt((math.pow(B, 2.0) + t_1)))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = -math.sqrt(-(F / A))
	elif t_4 <= -2e-202:
		tmp = math.sqrt((2.0 * (((math.pow(B, 2.0) - t_2) * F) * t_5))) / (t_2 - math.pow(B, 2.0))
	elif t_4 <= 0.0:
		tmp = -math.sqrt(-(F / C))
	elif t_4 <= 2e+277:
		tmp = -math.sqrt((t_0 * (t_5 * (2.0 * F)))) / t_0
	elif t_4 <= math.inf:
		tmp = 1.0 / (((A * (C * 4.0)) - math.pow(B, 2.0)) / (C * math.sqrt(((F * A) * -16.0))))
	else:
		tmp = -math.sqrt((2.0 * (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((B ^ 2.0) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(A - C) ^ 2.0
	t_2 = Float64(A * Float64(4.0 * C))
	t_3 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64(t_1 + (B ^ 2.0))))))) / t_3)
	t_5 = Float64(A + Float64(C + sqrt(Float64((B ^ 2.0) + t_1))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(-Float64(F / A))));
	elseif (t_4 <= -2e-202)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(Float64((B ^ 2.0) - t_2) * F) * t_5))) / Float64(t_2 - (B ^ 2.0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(-sqrt(Float64(-Float64(F / C))));
	elseif (t_4 <= 2e+277)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(t_5 * Float64(2.0 * F))))) / t_0);
	elseif (t_4 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(Float64(A * Float64(C * 4.0)) - (B ^ 2.0)) / Float64(C * sqrt(Float64(Float64(F * A) * -16.0)))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * 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 = (B ^ 2.0) - (4.0 * (A * C));
	t_1 = (A - C) ^ 2.0;
	t_2 = A * (4.0 * C);
	t_3 = (B ^ 2.0) - ((4.0 * A) * C);
	t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((t_1 + (B ^ 2.0)))))) / t_3;
	t_5 = A + (C + sqrt(((B ^ 2.0) + t_1)));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = -sqrt(-(F / A));
	elseif (t_4 <= -2e-202)
		tmp = sqrt((2.0 * ((((B ^ 2.0) - t_2) * F) * t_5))) / (t_2 - (B ^ 2.0));
	elseif (t_4 <= 0.0)
		tmp = -sqrt(-(F / C));
	elseif (t_4 <= 2e+277)
		tmp = -sqrt((t_0 * (t_5 * (2.0 * F)))) / t_0;
	elseif (t_4 <= Inf)
		tmp = 1.0 / (((A * (C * 4.0)) - (B ^ 2.0)) / (C * sqrt(((F * A) * -16.0))));
	else
		tmp = -sqrt((2.0 * (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[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(t$95$1 + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(A + N[(C + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], (-N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision]), If[LessEqual[t$95$4, -2e-202], N[(N[Sqrt[N[(2.0 * N[(N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], (-N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision]), If[LessEqual[t$95$4, 2e+277], N[((-N[Sqrt[N[(t$95$0 * N[(t$95$5 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(1.0 / N[(N[(N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] / N[(C * N[Sqrt[N[(N[(F * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]]]]]]]]]

Derivation

1. Split input into 6 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

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      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-50 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 64.0	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 64.0	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around inf 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right)} \]

   4. Simplified 49.5

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

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{A}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{A}} \]
      exponential.json-simplify-20 [=>] 49.5	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 49.5	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   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))) < -2.0000000000000001e-202

   1. Initial program 1.3

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

   2. Simplified 1.3

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

      [Start] 1.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 1.3	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   if -2.0000000000000001e-202 < (/.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))) < -0.0

   1. Initial program 60.5

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

   2. Simplified 57.3

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

      [Start] 60.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 60.5	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 60.5	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 60.5	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right)} \]

   4. Simplified 46.2

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

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{C}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{C}} \]
      exponential.json-simplify-20 [=>] 46.2	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational.json-simplify-9 [=>] 46.2	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

   if -0.0 < (/.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))) < 2.00000000000000001e277

   1. Initial program 1.6

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

   2. Simplified 3.9

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

      [Start] 1.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 1.6	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 1.6	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 1.6	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   if 2.00000000000000001e277 < (/.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 63.8

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

   2. Simplified 64.0

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

      [Start] 63.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 63.8	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 63.8	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 63.8	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Applied egg-rr 63.8

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

   4. Applied egg-rr 63.8

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

   5. Taylor expanded in C around inf 64.0

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

   6. Simplified 37.3

      \[\leadsto \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\color{blue}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}} \]
      Proof

      [Start] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \left(\sqrt{2} \cdot \left(C \cdot \sqrt{-8}\right)\right)}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \color{blue}{\left(C \cdot \left(\sqrt{-8} \cdot \sqrt{2}\right)\right)}}} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{A \cdot F} \cdot \left(C \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-8}\right)}\right)}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\color{blue}{C \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{A \cdot F}\right)}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{A \cdot F}\right)}} \]
      exponential.json-simplify-20 [=>] 37.3	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \color{blue}{\sqrt{\left(A \cdot F\right) \cdot \left(-8 \cdot 2\right)}}}} \]
      rational.json-simplify-2 [=>] 37.3	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\color{blue}{\left(F \cdot A\right)} \cdot \left(-8 \cdot 2\right)}}} \]
      metadata-eval [=>] 37.3	\[ \frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\left(F \cdot A\right) \cdot \color{blue}{-16}}}} \]

   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)))

   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

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      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-50 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around inf 63.3

      \[\leadsto \frac{\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   4. Taylor expanded in A around 0 53.5

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

   5. Simplified 53.5

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

      [Start] 53.5	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      rational.json-simplify-2 [=>] 53.5	\[ \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 53.5	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      exponential.json-simplify-20 [=>] 53.5	\[ -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
      rational.json-simplify-2 [=>] 53.5	\[ -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 41.4

   \[\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:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \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 -2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \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 0:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \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 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\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 \infty:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.4
Cost	217172

\[\begin{array}{l} t_0 := {\left(A - C\right)}^{2}\\ t_1 := A \cdot \left(4 \cdot C\right)\\ t_2 := \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_1\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + t_0}\right)\right)\right)}}{t_1 - {B}^{2}}\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{t_0 + {B}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;t_4 \leq 5 \cdot 10^{+189}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 2
Error	46.5
Cost	41304

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := -\sqrt{-\frac{F}{A}}\\ t_2 := A \cdot \left(C \cdot 4\right) - {B}^{2}\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -1.55 \cdot 10^{+124}:\\ \;\;\;\;\frac{C \cdot \left(-\sqrt{\left(A \cdot F\right) \cdot -16}\right)}{t_2}\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -7.8 \cdot 10^{-305}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 6.2 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{\frac{t_2}{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{elif}\;A \leq 3.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{A}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \sqrt{F \cdot \left(C \cdot -16\right)}\\ \end{array} \]

Alternative 3
Error	48.2
Cost	34200

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := t_0 - {B}^{2}\\ t_2 := C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}\\ t_3 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ t_4 := \frac{t_2}{t_3}\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(A + \left(C + \left(-B\right)\right)\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right)}} \cdot t_1}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-159}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-89}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{B \cdot \sqrt{\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right) \cdot 2}}{t_3}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 4
Error	48.2
Cost	33944

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := t_0 - {B}^{2}\\ t_2 := C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}\\ t_3 := \frac{t_2}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(A + \left(C + \left(-B\right)\right)\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right)}} \cdot t_1}\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 10^{-90}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{B}{\sqrt{2}} \cdot \left(-\sqrt{\frac{\frac{1}{F}}{A + \sqrt{{B}^{2} + {A}^{2}}}}\right)}\\ \mathbf{elif}\;B \leq 1.22 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 5
Error	48.2
Cost	33688

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := t_0 - {B}^{2}\\ t_2 := C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}\\ t_3 := \frac{t_2}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{if}\;B \leq -9 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(A + \left(C + \left(-B\right)\right)\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right)}} \cdot t_1}\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-159}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}\right)\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 6
Error	47.9
Cost	21512

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := \frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt{2 \cdot \left(\left(A + \left(C + \left(-B\right)\right)\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot F\right)\right)}} \cdot \left(t_0 - {B}^{2}\right)}\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-286}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 7
Error	47.9
Cost	21384

\[\begin{array}{l} t_0 := \frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ t_1 := A \cdot \left(C \cdot 4\right)\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{t_1 - {B}^{2}}{\sqrt{2 \cdot \left(\left(A + \left(C + \left(-B\right)\right)\right) \cdot \left(\left({B}^{2} - t_1\right) \cdot F\right)\right)}}}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 8
Error	47.9
Cost	21256

\[\begin{array}{l} t_0 := \frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ t_1 := A \cdot \left(4 \cdot C\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_1\right) \cdot F\right) \cdot \left(A + \left(C + \left(-B\right)\right)\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-286}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 9
Error	48.0
Cost	20744

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{t_0 - {B}^{2}}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(C + \left(C + A\right)\right)\right) \cdot 2}}{B}\\ \mathbf{elif}\;B \leq -5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(A - B\right)\right)\right)}}{{B}^{2} - t_0}\\ \mathbf{elif}\;B \leq -7.6 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-287}:\\ \;\;\;\;\frac{1}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \left(\sqrt{F \cdot \left(C \cdot -16\right)} \cdot A\right)\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 10
Error	48.6
Cost	14484

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -1.16 \cdot 10^{-35}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -2.2 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 4.8 \cdot 10^{-283}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 1.9 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{A}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \cdot \sqrt{F \cdot \left(C \cdot -16\right)}\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 5.1 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{+244}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	47.8
Cost	14484

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -6.4 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -2.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{C \cdot \left(-\sqrt{\left(A \cdot F\right) \cdot -16}\right)}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -6.2 \cdot 10^{-305}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 2.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{t_0 - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{t_0}\\ \end{array} \]

Alternative 12
Error	46.9
Cost	14352

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{-35}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -2.5 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{-276}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \end{array} \]

Alternative 13
Error	48.4
Cost	7904

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ t_2 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{if}\;C \leq -2.8 \cdot 10^{-35}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 3.8 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 2.4 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 5.1 \cdot 10^{+198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	48.6
Cost	7904

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -7 \cdot 10^{-34}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -1.15 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 2.1 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{+198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 10^{+244}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	48.2
Cost	7508

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -1.05 \cdot 10^{-34}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -2.2 \cdot 10^{-258}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;C \leq 7.2 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 5.1 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	47.2
Cost	7312

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -1.45 \cdot 10^{-139}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -9.5 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 6.2 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	47.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq -1.56 \cdot 10^{-277}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 18
Error	55.1
Cost	6720

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

Reproduce

herbie shell --seed 2023074 
(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))))