ABCF->ab-angle a

Percentage Accurate: 18.8% → 47.7%

Time: 36.9s

Alternatives: 22

Speedup: 5.0×

Specification

Math

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

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Initial Program: 18.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Alternative 1: 47.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (sqrt F)))
        (t_2 (- (pow B 2.0) t_0))
        (t_3 (sqrt (+ C (+ A (hypot B (- A C))))))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_2))
        (t_5 (fma B B (* C (* A -4.0)))))
   (if (<= t_4 0.0)
     (/
      (* t_3 (* (sqrt 2.0) (* (sqrt (- (* B B) t_0)) t_1)))
      (- (* B B) (* 4.0 (* A C))))
     (if (<= t_4 INFINITY)
       (/ (- (sqrt (* 2.0 (* F t_5)))) (/ t_5 t_3))
       (* (/ (sqrt 2.0) B) (* (sqrt (+ C (hypot C B))) t_1))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(F);
	double t_2 = pow(B, 2.0) - t_0;
	double t_3 = sqrt((C + (A + hypot(B, (A - C)))));
	double t_4 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_5 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = (t_3 * (sqrt(2.0) * (sqrt(((B * B) - t_0)) * t_1))) / ((B * B) - (4.0 * (A * C)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -sqrt((2.0 * (F * t_5))) / (t_5 / t_3);
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * t_1);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(F))
	t_2 = Float64((B ^ 2.0) - t_0)
	t_3 = sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C)))))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	t_5 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(Float64(t_3 * Float64(sqrt(2.0) * Float64(sqrt(Float64(Float64(B * B) - t_0)) * t_1))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * t_5)))) / Float64(t_5 / t_3));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(C, B))) * t_1));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[(t$95$3 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(N[(B * B), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[((-N[Sqrt[N[(2.0 * N[(F * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$5 / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 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))) < -0.0

   1. Initial program 30.5%

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

      1. associate-*l* 30.4%

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

      2. unpow2 30.4%

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

      3. +-commutative 30.4%

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

      4. unpow2 30.4%

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

      5. associate-*l* 30.4%

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

      6. unpow2 30.4%

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

   3. Simplified 30.4%

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

      1. sqrt-prod 35.1%

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

      2. *-commutative 35.1%

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

      3. *-commutative 35.1%

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

      4. associate-+l+ 35.9%

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

      5. unpow2 35.9%

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

      6. hypot-udef 49.5%

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

      7. associate-+r+ 48.2%

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

      8. +-commutative 48.2%

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

      9. associate-+r+ 48.9%

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

   5. Applied egg-rr 48.9%

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

      1. sqrt-prod 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. sqrt-prod 65.7%

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

      2. *-commutative 65.7%

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

      3. associate-*r* 65.7%

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

   9. Applied egg-rr 65.7%

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

   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))) < +inf.0

   1. Initial program 51.0%

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

      1. associate-*l* 51.0%

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

      2. unpow2 51.0%

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

      3. +-commutative 51.0%

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

      4. unpow2 51.0%

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

      5. associate-*l* 51.0%

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

      6. unpow2 51.0%

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

   3. Simplified 51.0%

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

      1. sqrt-prod 47.2%

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

      2. *-commutative 47.2%

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

      3. *-commutative 47.2%

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

      4. associate-+l+ 47.2%

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

      5. unpow2 47.2%

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

      6. hypot-udef 71.1%

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

      7. associate-+r+ 71.1%

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

      8. +-commutative 71.1%

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

      9. associate-+r+ 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. sqrt-prod 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. div-inv 71.0%

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

      2. distribute-rgt-neg-in 71.0%

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

      3. sqrt-unprod 71.2%

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

      4. *-commutative 71.2%

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

      5. associate-*r* 71.2%

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

      6. associate-*r* 71.2%

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

   9. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\left(\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)\right) \cdot \frac{1}{B \cdot B - \left(4 \cdot A\right) \cdot C}} \]
   10. Step-by-step derivation

       1. associate-*r/ 71.1%

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

       2. *-rgt-identity 71.1%

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

       3. distribute-rgt-neg-out 71.1%

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

       4. distribute-neg-frac 71.1%

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

       5. associate-/l* 71.2%

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

       6. distribute-neg-frac 71.2%

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

   11. Simplified 71.2%

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

   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)))

   1. Initial program 0.0%

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

   3. Simplified 0.4%

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

   4. Taylor expanded in A around 0 1.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 1.7%

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

      2. *-commutative 1.7%

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

      3. unpow2 1.7%

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

      4. unpow2 1.7%

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

   6. Simplified 1.7%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 1.7%

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

      2. hypot-def 30.7%

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

   8. Applied egg-rr 30.7%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 1.7%

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

      2. unpow2 1.7%

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

      3. unpow2 1.7%

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

      4. +-commutative 1.7%

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

      5. unpow2 1.7%

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

      6. unpow2 1.7%

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

      7. hypot-def 30.7%

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

   10. Simplified 30.7%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2} \cdot \left(\sqrt{B \cdot B - \left(4 \cdot A\right) \cdot C} \cdot \left(-\sqrt{F}\right)\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2: 45.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -1.6e-49)
     (/
      (* (sqrt (+ C (+ A (hypot B (- A C))))) (* (sqrt 2.0) (* B (sqrt F))))
      t_0)
     (if (<= B 1.02e+62)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (* (/ (sqrt 2.0) B) (* (sqrt (+ C (hypot C B))) (- (sqrt F))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.6e-49) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * (sqrt(2.0) * (B * sqrt(F)))) / t_0;
	} else if (B <= 1.02e+62) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.6e-49) {
		tmp = (Math.sqrt((C + (A + Math.hypot(B, (A - C))))) * (Math.sqrt(2.0) * (B * Math.sqrt(F)))) / t_0;
	} else if (B <= 1.02e+62) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt((C + Math.hypot(C, B))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -1.6e-49:
		tmp = (math.sqrt((C + (A + math.hypot(B, (A - C))))) * (math.sqrt(2.0) * (B * math.sqrt(F)))) / t_0
	elif B <= 1.02e+62:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt((C + math.hypot(C, B))) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -1.6e-49)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(sqrt(2.0) * Float64(B * sqrt(F)))) / t_0);
	elseif (B <= 1.02e+62)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(C, B))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -1.6e-49)
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * (sqrt(2.0) * (B * sqrt(F)))) / t_0;
	elseif (B <= 1.02e+62)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.6e-49], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(B * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1.02e+62], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.60000000000000001e-49

   1. Initial program 19.6%

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

      1. associate-*l* 19.6%

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

      2. unpow2 19.6%

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

      3. +-commutative 19.6%

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

      4. unpow2 19.6%

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

      5. associate-*l* 19.6%

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

      6. unpow2 19.6%

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

   3. Simplified 19.6%

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

      1. sqrt-prod 21.8%

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

      2. *-commutative 21.8%

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

      3. *-commutative 21.8%

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

      4. associate-+l+ 21.8%

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

      5. unpow2 21.8%

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

      6. hypot-udef 25.6%

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

      7. associate-+r+ 25.3%

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

      8. +-commutative 25.3%

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

      9. associate-+r+ 25.2%

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

   5. Applied egg-rr 25.2%

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

      1. sqrt-prod 25.2%

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

   7. Applied egg-rr 25.2%

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

   8. Taylor expanded in B around -inf 30.5%

      \[\leadsto \frac{-\left(\sqrt{2} \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{F} \cdot B\right)\right)}\right) \cdot \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 30.5%

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

      2. *-commutative 30.5%

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

      3. distribute-rgt-neg-in 30.5%

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

   10. Simplified 30.5%

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

   if -1.60000000000000001e-49 < B < 1.02000000000000002e62

   1. Initial program 22.2%

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

      1. associate-*l* 22.2%

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

      2. unpow2 22.2%

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

      3. +-commutative 22.2%

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

      4. unpow2 22.2%

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

      5. associate-*l* 22.2%

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

      6. unpow2 22.2%

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

   3. Simplified 22.2%

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

      1. sqrt-prod 22.7%

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

      2. *-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-+l+ 23.5%

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

      5. unpow2 23.5%

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

      6. hypot-udef 37.1%

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

      7. associate-+r+ 36.1%

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

      8. +-commutative 36.1%

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

      9. associate-+r+ 36.9%

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

   5. Applied egg-rr 36.9%

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

   6. Taylor expanded in A around -inf 20.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.1%

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

   8. Simplified 20.1%

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

   if 1.02000000000000002e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 3: 45.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -1.3e-42)
     (/
      (* (sqrt (+ C (+ A (hypot B (- A C))))) (* (sqrt F) (* B (sqrt 2.0))))
      t_0)
     (if (<= B 1e+62)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (* (/ (sqrt 2.0) B) (* (sqrt (+ C (hypot C B))) (- (sqrt F))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.3e-42) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * (sqrt(F) * (B * sqrt(2.0)))) / t_0;
	} else if (B <= 1e+62) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.3e-42) {
		tmp = (Math.sqrt((C + (A + Math.hypot(B, (A - C))))) * (Math.sqrt(F) * (B * Math.sqrt(2.0)))) / t_0;
	} else if (B <= 1e+62) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt((C + Math.hypot(C, B))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -1.3e-42:
		tmp = (math.sqrt((C + (A + math.hypot(B, (A - C))))) * (math.sqrt(F) * (B * math.sqrt(2.0)))) / t_0
	elif B <= 1e+62:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt((C + math.hypot(C, B))) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -1.3e-42)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(sqrt(F) * Float64(B * sqrt(2.0)))) / t_0);
	elseif (B <= 1e+62)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(C, B))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -1.3e-42)
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * (sqrt(F) * (B * sqrt(2.0)))) / t_0;
	elseif (B <= 1e+62)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.3e-42], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(B * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1e+62], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.3e-42

   1. Initial program 19.6%

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

      1. associate-*l* 19.6%

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

      2. unpow2 19.6%

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

      3. +-commutative 19.6%

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

      4. unpow2 19.6%

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

      5. associate-*l* 19.6%

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

      6. unpow2 19.6%

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

   3. Simplified 19.6%

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

      1. sqrt-prod 21.8%

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

      2. *-commutative 21.8%

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

      3. *-commutative 21.8%

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

      4. associate-+l+ 21.8%

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

      5. unpow2 21.8%

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

      6. hypot-udef 25.6%

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

      7. associate-+r+ 25.3%

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

      8. +-commutative 25.3%

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

      9. associate-+r+ 25.2%

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

   5. Applied egg-rr 25.2%

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

   6. Taylor expanded in B around -inf 30.5%

      \[\leadsto \frac{-\color{blue}{\left(-1 \cdot \left(\left(\sqrt{2} \cdot B\right) \cdot \sqrt{F}\right)\right)} \cdot \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 30.5%

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

      2. *-commutative 30.5%

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

      3. neg-mul-1 30.5%

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

      4. distribute-lft-neg-in 30.5%

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

   8. Simplified 30.5%

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

   if -1.3e-42 < B < 1.00000000000000004e62

   1. Initial program 22.2%

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

      1. associate-*l* 22.2%

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

      2. unpow2 22.2%

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

      3. +-commutative 22.2%

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

      4. unpow2 22.2%

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

      5. associate-*l* 22.2%

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

      6. unpow2 22.2%

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

   3. Simplified 22.2%

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

      1. sqrt-prod 22.7%

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

      2. *-commutative 22.7%

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

      3. *-commutative 22.7%

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

      4. associate-+l+ 23.5%

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

      5. unpow2 23.5%

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

      6. hypot-udef 37.1%

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

      7. associate-+r+ 36.1%

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

      8. +-commutative 36.1%

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

      9. associate-+r+ 36.9%

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

   5. Applied egg-rr 36.9%

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

   6. Taylor expanded in A around -inf 20.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.1%

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

   8. Simplified 20.1%

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

   if 1.00000000000000004e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(B \cdot \sqrt{2}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 4: 45.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(F \cdot t_1\right)\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{t_2}\right) \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-190}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (- (* B B) (* (* 4.0 A) C)))
        (t_2 (* 2.0 (* F t_1))))
   (if (<= B -4.5e+146)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.25e-120)
       (* (* (sqrt (+ C (+ A (hypot B (- A C))))) (sqrt t_2)) (/ -1.0 t_1))
       (if (<= B -7e-264)
         (/ (- (sqrt (* t_2 (fma 2.0 C (/ -0.5 (/ A (* B B))))))) t_1)
         (if (<= B 6.2e-224)
           (/
            (*
             (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
             (- (sqrt (* 2.0 C))))
            t_0)
           (if (<= B 1.65e-190)
             (*
              -0.5
              (*
               (sqrt 2.0)
               (sqrt (/ F (/ A (fma 0.5 (/ (* B B) (* A A)) -2.0))))))
             (if (<= B 1e+62)
               (/
                (*
                 (sqrt (* 2.0 (* F t_0)))
                 (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
                t_0)
               (*
                (/ (sqrt 2.0) B)
                (* (sqrt (+ C (hypot C B))) (- (sqrt F))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) - ((4.0 * A) * C);
	double t_2 = 2.0 * (F * t_1);
	double tmp;
	if (B <= -4.5e+146) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.25e-120) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * sqrt(t_2)) * (-1.0 / t_1);
	} else if (B <= -7e-264) {
		tmp = -sqrt((t_2 * fma(2.0, C, (-0.5 / (A / (B * B)))))) / t_1;
	} else if (B <= 6.2e-224) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_0;
	} else if (B <= 1.65e-190) {
		tmp = -0.5 * (sqrt(2.0) * sqrt((F / (A / fma(0.5, ((B * B) / (A * A)), -2.0)))));
	} else if (B <= 1e+62) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * -sqrt(F));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(2.0 * Float64(F * t_1))
	tmp = 0.0
	if (B <= -4.5e+146)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.25e-120)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * sqrt(t_2)) * Float64(-1.0 / t_1));
	elseif (B <= -7e-264)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * fma(2.0, C, Float64(-0.5 / Float64(A / Float64(B * B))))))) / t_1);
	elseif (B <= 6.2e-224)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	elseif (B <= 1.65e-190)
		tmp = Float64(-0.5 * Float64(sqrt(2.0) * sqrt(Float64(F / Float64(A / fma(0.5, Float64(Float64(B * B) / Float64(A * A)), -2.0))))));
	elseif (B <= 1e+62)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(C, B))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4.5e+146], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.25e-120], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7e-264], N[((-N[Sqrt[N[(t$95$2 * N[(2.0 * C + N[(-0.5 / N[(A / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 6.2e-224], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 1.65e-190], N[(-0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / N[(A / N[(0.5 * N[(N[(B * B), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1e+62], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.50000000000000026e146

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 2.4%

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

   5. Taylor expanded in B around -inf 8.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.1%

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

   7. Simplified 8.1%

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

   if -4.50000000000000026e146 < B < -1.25000000000000002e-120

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. unpow2 36.9%

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

      3. +-commutative 36.9%

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

      4. unpow2 36.9%

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

      5. associate-*l* 36.9%

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

      6. unpow2 36.9%

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

   3. Simplified 36.9%

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

      1. sqrt-prod 42.0%

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

      2. *-commutative 42.0%

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

      3. *-commutative 42.0%

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

      4. associate-+l+ 42.3%

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

      5. unpow2 42.3%

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

      6. hypot-udef 49.7%

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

      7. associate-+r+ 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-+r+ 48.9%

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

   5. Applied egg-rr 48.9%

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

      1. sqrt-prod 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. div-inv 48.9%

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

      2. distribute-rgt-neg-in 48.9%

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

      3. sqrt-unprod 48.9%

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

      4. *-commutative 48.9%

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

      5. associate-*r* 48.9%

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

      6. associate-*r* 48.9%

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

   9. Applied egg-rr 48.9%

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

   if -1.25000000000000002e-120 < B < -7.0000000000000001e-264

   1. Initial program 39.6%

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

      1. associate-*l* 39.2%

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

      2. unpow2 39.2%

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

      3. +-commutative 39.2%

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

      4. unpow2 39.2%

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

      5. associate-*l* 39.2%

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

      6. unpow2 39.2%

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

   3. Simplified 39.2%

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

      1. unpow2 39.2%

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

      2. hypot-udef 43.6%

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

      3. *-un-lft-identity 43.6%

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

   5. Applied egg-rr 43.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in A around -inf 23.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. fma-def 23.7%

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

      2. associate-*r/ 23.7%

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

      3. unpow2 23.7%

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

   10. Simplified 23.7%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   11. Step-by-step derivation

       1. distribute-frac-neg 23.7%

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

       2. *-commutative 23.7%

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

       3. *-commutative 23.7%

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

       4. *-commutative 23.7%

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

       5. associate-*r* 23.7%

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

       6. associate-/l* 23.7%

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

   12. Applied egg-rr 24.2%

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

   if -7.0000000000000001e-264 < B < 6.20000000000000017e-224

   1. Initial program 6.6%

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

      1. associate-*l* 6.6%

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

      2. unpow2 6.6%

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

      3. +-commutative 6.6%

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

      4. unpow2 6.6%

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

      5. associate-*l* 6.6%

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

      6. unpow2 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 11.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 24.6%

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

      2. *-commutative 24.6%

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

      3. *-commutative 24.6%

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

   6. Applied egg-rr 24.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 24.6%

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

      2. unpow2 24.6%

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

      3. cancel-sign-sub-inv 24.6%

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

      4. unpow2 24.6%

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

      5. metadata-eval 24.6%

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

   8. Simplified 24.6%

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

   if 6.20000000000000017e-224 < B < 1.65000000000000009e-190

   1. Initial program 10.7%

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

      1. associate-*l* 10.7%

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

      2. unpow2 10.7%

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

      3. +-commutative 10.7%

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

      4. unpow2 10.7%

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

      5. associate-*l* 10.7%

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

      6. unpow2 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in A around -inf 2.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + \left(-0.5 \cdot \frac{{B}^{2}}{A} + -1 \cdot A\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \color{blue}{\left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) + -1 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. mul-1-neg 2.4%

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

      3. unsub-neg 2.4%

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

      4. distribute-lft-out 2.4%

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

      5. *-commutative 2.4%

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

      6. unpow2 2.4%

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

      7. times-frac 1.7%

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

      8. unpow2 1.7%

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

      9. unpow2 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in C around -inf 18.5%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{F \cdot \left(0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2\right)}{A}} \cdot \sqrt{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 18.5%

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

      2. associate-/l* 18.5%

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

      3. fma-neg 18.5%

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

      4. unpow2 18.5%

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

      5. unpow2 18.5%

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

      6. metadata-eval 18.5%

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

   9. Simplified 18.5%

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

   if 1.65000000000000009e-190 < B < 1.00000000000000004e62

   1. Initial program 21.1%

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

      1. associate-*l* 21.1%

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

      2. unpow2 21.1%

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

      3. +-commutative 21.1%

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

      4. unpow2 21.1%

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

      5. associate-*l* 21.1%

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

      6. unpow2 21.1%

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

   3. Simplified 21.1%

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

      1. sqrt-prod 22.4%

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

      2. *-commutative 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-+l+ 22.4%

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

      5. unpow2 22.4%

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

      6. hypot-udef 37.8%

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

      7. associate-+r+ 37.1%

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

      8. +-commutative 37.1%

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

      9. associate-+r+ 38.3%

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

   5. Applied egg-rr 38.3%

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

   6. Taylor expanded in A around -inf 20.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.4%

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

   8. Simplified 20.4%

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

   if 1.00000000000000004e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)}\right) \cdot \frac{-1}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-190}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(F \cdot t_1\right)\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{t_2}\right) \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-191}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (- (* B B) (* (* 4.0 A) C)))
        (t_2 (* 2.0 (* F t_1))))
   (if (<= B -4.2e+146)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.95e-120)
       (* (* (sqrt (+ C (+ A (hypot B (- A C))))) (sqrt t_2)) (/ -1.0 t_1))
       (if (<= B -1.25e-260)
         (/ (- (sqrt (* t_2 (fma 2.0 C (/ -0.5 (/ A (* B B))))))) t_1)
         (if (<= B 6.8e-224)
           (/
            (*
             (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
             (- (sqrt (* 2.0 C))))
            t_0)
           (if (<= B 7.5e-191)
             (*
              -0.5
              (*
               (sqrt 2.0)
               (sqrt (/ F (/ A (fma 0.5 (/ (* B B) (* A A)) -2.0))))))
             (if (<= B 6.5e+62)
               (/
                (*
                 (sqrt (* 2.0 (* F t_0)))
                 (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
                t_0)
               (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt B))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = (B * B) - ((4.0 * A) * C);
	double t_2 = 2.0 * (F * t_1);
	double tmp;
	if (B <= -4.2e+146) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.95e-120) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * sqrt(t_2)) * (-1.0 / t_1);
	} else if (B <= -1.25e-260) {
		tmp = -sqrt((t_2 * fma(2.0, C, (-0.5 / (A / (B * B)))))) / t_1;
	} else if (B <= 6.8e-224) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_0;
	} else if (B <= 7.5e-191) {
		tmp = -0.5 * (sqrt(2.0) * sqrt((F / (A / fma(0.5, ((B * B) / (A * A)), -2.0)))));
	} else if (B <= 6.5e+62) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(2.0 * Float64(F * t_1))
	tmp = 0.0
	if (B <= -4.2e+146)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.95e-120)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * sqrt(t_2)) * Float64(-1.0 / t_1));
	elseif (B <= -1.25e-260)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * fma(2.0, C, Float64(-0.5 / Float64(A / Float64(B * B))))))) / t_1);
	elseif (B <= 6.8e-224)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	elseif (B <= 7.5e-191)
		tmp = Float64(-0.5 * Float64(sqrt(2.0) * sqrt(Float64(F / Float64(A / fma(0.5, Float64(Float64(B * B) / Float64(A * A)), -2.0))))));
	elseif (B <= 6.5e+62)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(B))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4.2e+146], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.95e-120], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.25e-260], N[((-N[Sqrt[N[(t$95$2 * N[(2.0 * C + N[(-0.5 / N[(A / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 6.8e-224], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 7.5e-191], N[(-0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / N[(A / N[(0.5 * N[(N[(B * B), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.5e+62], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.2000000000000001e146

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 2.4%

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

   5. Taylor expanded in B around -inf 8.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.1%

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

   7. Simplified 8.1%

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

   if -4.2000000000000001e146 < B < -1.9500000000000001e-120

   1. Initial program 36.9%

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

      1. associate-*l* 36.9%

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

      2. unpow2 36.9%

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

      3. +-commutative 36.9%

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

      4. unpow2 36.9%

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

      5. associate-*l* 36.9%

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

      6. unpow2 36.9%

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

   3. Simplified 36.9%

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

      1. sqrt-prod 42.0%

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

      2. *-commutative 42.0%

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

      3. *-commutative 42.0%

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

      4. associate-+l+ 42.3%

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

      5. unpow2 42.3%

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

      6. hypot-udef 49.7%

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

      7. associate-+r+ 49.1%

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

      8. +-commutative 49.1%

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

      9. associate-+r+ 48.9%

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

   5. Applied egg-rr 48.9%

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

      1. sqrt-prod 48.9%

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

   7. Applied egg-rr 48.9%

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

      1. div-inv 48.9%

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

      2. distribute-rgt-neg-in 48.9%

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

      3. sqrt-unprod 48.9%

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

      4. *-commutative 48.9%

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

      5. associate-*r* 48.9%

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

      6. associate-*r* 48.9%

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

   9. Applied egg-rr 48.9%

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

   if -1.9500000000000001e-120 < B < -1.2500000000000001e-260

   1. Initial program 39.6%

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

      1. associate-*l* 39.2%

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

      2. unpow2 39.2%

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

      3. +-commutative 39.2%

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

      4. unpow2 39.2%

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

      5. associate-*l* 39.2%

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

      6. unpow2 39.2%

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

   3. Simplified 39.2%

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

      1. unpow2 39.2%

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

      2. hypot-udef 43.6%

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

      3. *-un-lft-identity 43.6%

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

   5. Applied egg-rr 43.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in A around -inf 23.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. fma-def 23.7%

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

      2. associate-*r/ 23.7%

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

      3. unpow2 23.7%

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

   10. Simplified 23.7%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   11. Step-by-step derivation

       1. distribute-frac-neg 23.7%

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

       2. *-commutative 23.7%

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

       3. *-commutative 23.7%

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

       4. *-commutative 23.7%

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

       5. associate-*r* 23.7%

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

       6. associate-/l* 23.7%

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

   12. Applied egg-rr 24.2%

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

   if -1.2500000000000001e-260 < B < 6.79999999999999984e-224

   1. Initial program 6.6%

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

      1. associate-*l* 6.6%

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

      2. unpow2 6.6%

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

      3. +-commutative 6.6%

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

      4. unpow2 6.6%

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

      5. associate-*l* 6.6%

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

      6. unpow2 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 11.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 24.6%

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

      2. *-commutative 24.6%

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

      3. *-commutative 24.6%

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

   6. Applied egg-rr 24.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 24.6%

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

      2. unpow2 24.6%

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

      3. cancel-sign-sub-inv 24.6%

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

      4. unpow2 24.6%

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

      5. metadata-eval 24.6%

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

   8. Simplified 24.6%

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

   if 6.79999999999999984e-224 < B < 7.4999999999999995e-191

   1. Initial program 10.7%

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

      1. associate-*l* 10.7%

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

      2. unpow2 10.7%

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

      3. +-commutative 10.7%

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

      4. unpow2 10.7%

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

      5. associate-*l* 10.7%

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

      6. unpow2 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in A around -inf 2.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + \left(-0.5 \cdot \frac{{B}^{2}}{A} + -1 \cdot A\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \color{blue}{\left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) + -1 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. mul-1-neg 2.4%

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

      3. unsub-neg 2.4%

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

      4. distribute-lft-out 2.4%

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

      5. *-commutative 2.4%

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

      6. unpow2 2.4%

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

      7. times-frac 1.7%

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

      8. unpow2 1.7%

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

      9. unpow2 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in C around -inf 18.5%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{F \cdot \left(0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2\right)}{A}} \cdot \sqrt{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 18.5%

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

      2. associate-/l* 18.5%

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

      3. fma-neg 18.5%

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

      4. unpow2 18.5%

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

      5. unpow2 18.5%

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

      6. metadata-eval 18.5%

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

   9. Simplified 18.5%

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

   if 7.4999999999999995e-191 < B < 6.5000000000000003e62

   1. Initial program 21.1%

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

      1. associate-*l* 21.1%

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

      2. unpow2 21.1%

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

      3. +-commutative 21.1%

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

      4. unpow2 21.1%

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

      5. associate-*l* 21.1%

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

      6. unpow2 21.1%

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

   3. Simplified 21.1%

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

      1. sqrt-prod 22.4%

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

      2. *-commutative 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-+l+ 22.4%

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

      5. unpow2 22.4%

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

      6. hypot-udef 37.8%

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

      7. associate-+r+ 37.1%

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

      8. +-commutative 37.1%

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

      9. associate-+r+ 38.3%

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

   5. Applied egg-rr 38.3%

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

   6. Taylor expanded in A around -inf 20.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.4%

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

   8. Simplified 20.4%

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

   if 6.5000000000000003e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

   11. Taylor expanded in C around 0 74.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.2 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-120}:\\ \;\;\;\;\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)}\right) \cdot \frac{-1}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-191}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 6: 43.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ t_2 := B \cdot B - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-121}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{t_2}\\ \mathbf{elif}\;B \leq 6.3 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-191}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1 (sqrt (* 2.0 (* F t_0))))
        (t_2 (- (* B B) (* (* 4.0 A) C))))
   (if (<= B -4.5e+151)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -9e-121)
       (/ (* t_1 (- (sqrt (+ C (+ A (hypot B (- A C))))))) t_0)
       (if (<= B -1.12e-264)
         (/
          (- (sqrt (* (* 2.0 (* F t_2)) (fma 2.0 C (/ -0.5 (/ A (* B B)))))))
          t_2)
         (if (<= B 6.3e-224)
           (/
            (*
             (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
             (- (sqrt (* 2.0 C))))
            t_0)
           (if (<= B 8.5e-191)
             (*
              -0.5
              (*
               (sqrt 2.0)
               (sqrt (/ F (/ A (fma 0.5 (/ (* B B) (* A A)) -2.0))))))
             (if (<= B 1e+62)
               (/ (* t_1 (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A))))))) t_0)
               (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt B))))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = sqrt((2.0 * (F * t_0)));
	double t_2 = (B * B) - ((4.0 * A) * C);
	double tmp;
	if (B <= -4.5e+151) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -9e-121) {
		tmp = (t_1 * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	} else if (B <= -1.12e-264) {
		tmp = -sqrt(((2.0 * (F * t_2)) * fma(2.0, C, (-0.5 / (A / (B * B)))))) / t_2;
	} else if (B <= 6.3e-224) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_0;
	} else if (B <= 8.5e-191) {
		tmp = -0.5 * (sqrt(2.0) * sqrt((F / (A / fma(0.5, ((B * B) / (A * A)), -2.0)))));
	} else if (B <= 1e+62) {
		tmp = (t_1 * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * Float64(F * t_0)))
	t_2 = Float64(Float64(B * B) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B <= -4.5e+151)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -9e-121)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0);
	elseif (B <= -1.12e-264)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_2)) * fma(2.0, C, Float64(-0.5 / Float64(A / Float64(B * B))))))) / t_2);
	elseif (B <= 6.3e-224)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	elseif (B <= 8.5e-191)
		tmp = Float64(-0.5 * Float64(sqrt(2.0) * sqrt(Float64(F / Float64(A / fma(0.5, Float64(Float64(B * B) / Float64(A * A)), -2.0))))));
	elseif (B <= 1e+62)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(B))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4.5e+151], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -9e-121], N[(N[(t$95$1 * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, -1.12e-264], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 / N[(A / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 6.3e-224], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 8.5e-191], N[(-0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / N[(A / N[(0.5 * N[(N[(B * B), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1e+62], N[(N[(t$95$1 * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -4.4999999999999999e151

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 5.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.9%

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

   7. Simplified 5.9%

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

   if -4.4999999999999999e151 < B < -9.0000000000000007e-121

   1. Initial program 36.3%

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

      1. associate-*l* 36.3%

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

      2. unpow2 36.3%

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

      3. +-commutative 36.3%

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

      4. unpow2 36.3%

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

      5. associate-*l* 36.3%

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

      6. unpow2 36.3%

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

   3. Simplified 36.3%

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

      1. sqrt-prod 41.3%

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

      2. *-commutative 41.3%

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

      3. *-commutative 41.3%

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

      4. associate-+l+ 41.6%

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

      5. unpow2 41.6%

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

      6. hypot-udef 50.6%

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

      7. associate-+r+ 50.0%

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

      8. +-commutative 50.0%

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

      9. associate-+r+ 49.7%

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

   5. Applied egg-rr 49.7%

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

   if -9.0000000000000007e-121 < B < -1.12000000000000009e-264

   1. Initial program 39.6%

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

      1. associate-*l* 39.2%

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

      2. unpow2 39.2%

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

      3. +-commutative 39.2%

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

      4. unpow2 39.2%

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

      5. associate-*l* 39.2%

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

      6. unpow2 39.2%

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

   3. Simplified 39.2%

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

      1. unpow2 39.2%

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

      2. hypot-udef 43.6%

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

      3. *-un-lft-identity 43.6%

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

   5. Applied egg-rr 43.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in A around -inf 23.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. fma-def 23.7%

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

      2. associate-*r/ 23.7%

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

      3. unpow2 23.7%

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

   10. Simplified 23.7%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   11. Step-by-step derivation

       1. distribute-frac-neg 23.7%

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

       2. *-commutative 23.7%

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

       3. *-commutative 23.7%

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

       4. *-commutative 23.7%

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

       5. associate-*r* 23.7%

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

       6. associate-/l* 23.7%

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

   12. Applied egg-rr 24.2%

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

   if -1.12000000000000009e-264 < B < 6.30000000000000043e-224

   1. Initial program 6.6%

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

      1. associate-*l* 6.6%

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

      2. unpow2 6.6%

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

      3. +-commutative 6.6%

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

      4. unpow2 6.6%

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

      5. associate-*l* 6.6%

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

      6. unpow2 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 11.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 24.6%

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

      2. *-commutative 24.6%

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

      3. *-commutative 24.6%

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

   6. Applied egg-rr 24.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 24.6%

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

      2. unpow2 24.6%

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

      3. cancel-sign-sub-inv 24.6%

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

      4. unpow2 24.6%

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

      5. metadata-eval 24.6%

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

   8. Simplified 24.6%

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

   if 6.30000000000000043e-224 < B < 8.49999999999999954e-191

   1. Initial program 10.7%

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

      1. associate-*l* 10.7%

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

      2. unpow2 10.7%

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

      3. +-commutative 10.7%

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

      4. unpow2 10.7%

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

      5. associate-*l* 10.7%

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

      6. unpow2 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in A around -inf 2.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + \left(-0.5 \cdot \frac{{B}^{2}}{A} + -1 \cdot A\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. associate-+r+ 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \color{blue}{\left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) + -1 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. mul-1-neg 2.4%

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

      3. unsub-neg 2.4%

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

      4. distribute-lft-out 2.4%

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

      5. *-commutative 2.4%

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

      6. unpow2 2.4%

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

      7. times-frac 1.7%

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

      8. unpow2 1.7%

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

      9. unpow2 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in C around -inf 18.5%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{F \cdot \left(0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2\right)}{A}} \cdot \sqrt{2}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 18.5%

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

      2. associate-/l* 18.5%

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

      3. fma-neg 18.5%

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

      4. unpow2 18.5%

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

      5. unpow2 18.5%

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

      6. metadata-eval 18.5%

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

   9. Simplified 18.5%

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

   if 8.49999999999999954e-191 < B < 1.00000000000000004e62

   1. Initial program 21.1%

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

      1. associate-*l* 21.1%

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

      2. unpow2 21.1%

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

      3. +-commutative 21.1%

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

      4. unpow2 21.1%

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

      5. associate-*l* 21.1%

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

      6. unpow2 21.1%

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

   3. Simplified 21.1%

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

      1. sqrt-prod 22.4%

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

      2. *-commutative 22.4%

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

      3. *-commutative 22.4%

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

      4. associate-+l+ 22.4%

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

      5. unpow2 22.4%

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

      6. hypot-udef 37.8%

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

      7. associate-+r+ 37.1%

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

      8. +-commutative 37.1%

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

      9. associate-+r+ 38.3%

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

   5. Applied egg-rr 38.3%

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

   6. Taylor expanded in A around -inf 20.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.4%

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

   8. Simplified 20.4%

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

   if 1.00000000000000004e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

   11. Taylor expanded in C around 0 74.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B \cdot B}}\right)}}{B \cdot B - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 6.3 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-191}:\\ \;\;\;\;-0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 7: 42.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot t_1\right)} \cdot \frac{-1}{t_0}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_1} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* F t_0)))
   (if (<= B -4e+146)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.1e-98)
       (* (sqrt (* 2.0 (* (+ C (+ A (hypot B (- A C)))) t_1))) (/ -1.0 t_0))
       (if (<= B 2.05e+62)
         (/
          (* (sqrt (* 2.0 t_1)) (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
          t_0)
         (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt B)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = F * t_0;
	double tmp;
	if (B <= -4e+146) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.1e-98) {
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * t_1))) * (-1.0 / t_0);
	} else if (B <= 2.05e+62) {
		tmp = (sqrt((2.0 * t_1)) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = F * t_0;
	double tmp;
	if (B <= -4e+146) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.1e-98) {
		tmp = Math.sqrt((2.0 * ((C + (A + Math.hypot(B, (A - C)))) * t_1))) * (-1.0 / t_0);
	} else if (B <= 2.05e+62) {
		tmp = (Math.sqrt((2.0 * t_1)) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(F) * -Math.sqrt(B));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = F * t_0
	tmp = 0
	if B <= -4e+146:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.1e-98:
		tmp = math.sqrt((2.0 * ((C + (A + math.hypot(B, (A - C)))) * t_1))) * (-1.0 / t_0)
	elif B <= 2.05e+62:
		tmp = (math.sqrt((2.0 * t_1)) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(F) * -math.sqrt(B))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(F * t_0)
	tmp = 0.0
	if (B <= -4e+146)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.1e-98)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(C + Float64(A + hypot(B, Float64(A - C)))) * t_1))) * Float64(-1.0 / t_0));
	elseif (B <= 2.05e+62)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(B))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = F * t_0;
	tmp = 0.0;
	if (B <= -4e+146)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.1e-98)
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * t_1))) * (-1.0 / t_0);
	elseif (B <= 2.05e+62)
		tmp = (sqrt((2.0 * t_1)) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[B, -4e+146], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.1e-98], N[(N[Sqrt[N[(2.0 * N[(N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.05e+62], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -3.99999999999999973e146

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 2.4%

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

   5. Taylor expanded in B around -inf 8.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.1%

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

   7. Simplified 8.1%

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

   if -3.99999999999999973e146 < B < -1.09999999999999998e-98

   1. Initial program 40.4%

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

      1. associate-*l* 40.4%

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

      2. unpow2 40.4%

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

      3. +-commutative 40.4%

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

      4. unpow2 40.4%

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

      5. associate-*l* 40.4%

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

      6. unpow2 40.4%

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

   3. Simplified 40.4%

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

      1. div-inv 40.4%

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

   5. Applied egg-rr 46.4%

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

   if -1.09999999999999998e-98 < B < 2.04999999999999992e62

   1. Initial program 20.2%

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

      1. associate-*l* 20.1%

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

      2. unpow2 20.1%

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

      3. +-commutative 20.1%

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

      4. unpow2 20.1%

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

      5. associate-*l* 20.1%

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

      6. unpow2 20.1%

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

   3. Simplified 20.1%

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

      1. sqrt-prod 20.6%

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

      2. *-commutative 20.6%

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

      3. *-commutative 20.6%

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

      4. associate-+l+ 21.3%

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

      5. unpow2 21.3%

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

      6. hypot-udef 35.1%

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

      7. associate-+r+ 34.1%

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

      8. +-commutative 34.1%

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

      9. associate-+r+ 35.0%

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

   5. Applied egg-rr 35.0%

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

   6. Taylor expanded in A around -inf 20.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.6%

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

   8. Simplified 20.6%

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

   if 2.04999999999999992e62 < B

   1. Initial program 7.7%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in A around 0 13.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 13.4%

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

      2. *-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

   6. Simplified 13.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]
   7. Step-by-step derivation

      1. sqrt-prod 15.2%

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

      2. hypot-def 75.3%

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

   8. Applied egg-rr 75.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)} \]
   9. Step-by-step derivation

      1. hypot-def 15.2%

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

      2. unpow2 15.2%

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

      3. unpow2 15.2%

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

      4. +-commutative 15.2%

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

      5. unpow2 15.2%

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

      6. unpow2 15.2%

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

      7. hypot-def 75.3%

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

   10. Simplified 75.3%

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

   11. Taylor expanded in C around 0 74.0%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 8: 38.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_2 := F \cdot t_1\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot t_2\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_2} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (- (* B B) (* 4.0 (* A C)))) (t_2 (* F t_1)))
   (if (<= B -4.2e+144)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.7e-98)
       (* (sqrt (* 2.0 (* (+ C (+ A (hypot B (- A C)))) t_2))) (/ -1.0 t_1))
       (if (<= B 4e+80)
         (/
          (* (sqrt (* 2.0 t_2)) (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
          t_1)
         (if (<= B 3.6e+282)
           (* (sqrt (+ (* C F) (* B F))) (/ t_0 B))
           (* (sqrt (/ F B)) t_0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = F * t_1;
	double tmp;
	if (B <= -4.2e+144) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-98) {
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * t_2))) * (-1.0 / t_1);
	} else if (B <= 4e+80) {
		tmp = (sqrt((2.0 * t_2)) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	} else if (B <= 3.6e+282) {
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double t_2 = F * t_1;
	double tmp;
	if (B <= -4.2e+144) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-98) {
		tmp = Math.sqrt((2.0 * ((C + (A + Math.hypot(B, (A - C)))) * t_2))) * (-1.0 / t_1);
	} else if (B <= 4e+80) {
		tmp = (Math.sqrt((2.0 * t_2)) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	} else if (B <= 3.6e+282) {
		tmp = Math.sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = (B * B) - (4.0 * (A * C))
	t_2 = F * t_1
	tmp = 0
	if B <= -4.2e+144:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.7e-98:
		tmp = math.sqrt((2.0 * ((C + (A + math.hypot(B, (A - C)))) * t_2))) * (-1.0 / t_1)
	elif B <= 4e+80:
		tmp = (math.sqrt((2.0 * t_2)) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1
	elif B <= 3.6e+282:
		tmp = math.sqrt(((C * F) + (B * F))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_2 = Float64(F * t_1)
	tmp = 0.0
	if (B <= -4.2e+144)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.7e-98)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(C + Float64(A + hypot(B, Float64(A - C)))) * t_2))) * Float64(-1.0 / t_1));
	elseif (B <= 4e+80)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_2)) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_1);
	elseif (B <= 3.6e+282)
		tmp = Float64(sqrt(Float64(Float64(C * F) + Float64(B * F))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = (B * B) - (4.0 * (A * C));
	t_2 = F * t_1;
	tmp = 0.0;
	if (B <= -4.2e+144)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.7e-98)
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * t_2))) * (-1.0 / t_1);
	elseif (B <= 4e+80)
		tmp = (sqrt((2.0 * t_2)) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	elseif (B <= 3.6e+282)
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * t$95$1), $MachinePrecision]}, If[LessEqual[B, -4.2e+144], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.7e-98], N[(N[Sqrt[N[(2.0 * N[(N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4e+80], N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 3.6e+282], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] + N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -4.19999999999999993e144

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 2.4%

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

   5. Taylor expanded in B around -inf 8.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.1%

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

   7. Simplified 8.1%

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

   if -4.19999999999999993e144 < B < -1.7000000000000001e-98

   1. Initial program 40.4%

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

      1. associate-*l* 40.4%

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

      2. unpow2 40.4%

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

      3. +-commutative 40.4%

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

      4. unpow2 40.4%

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

      5. associate-*l* 40.4%

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

      6. unpow2 40.4%

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

   3. Simplified 40.4%

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

      1. div-inv 40.4%

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

   5. Applied egg-rr 46.4%

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

   if -1.7000000000000001e-98 < B < 4e80

   1. Initial program 19.9%

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

      1. associate-*l* 19.8%

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

      2. unpow2 19.8%

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

      3. +-commutative 19.8%

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

      4. unpow2 19.8%

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

      5. associate-*l* 19.8%

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

      6. unpow2 19.8%

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

   3. Simplified 19.8%

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

      1. sqrt-prod 20.3%

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

      2. *-commutative 20.3%

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

      3. *-commutative 20.3%

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

      4. associate-+l+ 21.0%

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

      5. unpow2 21.0%

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

      6. hypot-udef 34.6%

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

      7. associate-+r+ 33.6%

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

      8. +-commutative 33.6%

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

      9. associate-+r+ 34.4%

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

   5. Applied egg-rr 34.4%

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

   6. Taylor expanded in A around -inf 20.2%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. unpow2 20.2%

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

   8. Simplified 20.2%

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

   if 4e80 < B < 3.59999999999999986e282

   1. Initial program 9.7%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

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

      2. *-commutative 16.4%

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

      3. unpow2 16.4%

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

      4. unpow2 16.4%

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

   6. Simplified 16.4%

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

   7. Taylor expanded in C around 0 61.4%

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

   if 3.59999999999999986e282 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 0.0%

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

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

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

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.2 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9: 38.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot t_1\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+283}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -4e+146)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.02e-55)
       (*
        (sqrt (* 2.0 (* (+ C (+ A (hypot B (- A C)))) (* F t_1))))
        (/ -1.0 t_1))
       (if (<= B 1.25e+81)
         (/
          (*
           (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
           (- (sqrt (* 2.0 C))))
          t_1)
         (if (<= B 1.9e+283)
           (* (sqrt (+ (* C F) (* B F))) (/ t_0 B))
           (* (sqrt (/ F B)) t_0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -4e+146) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.02e-55) {
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * (F * t_1)))) * (-1.0 / t_1);
	} else if (B <= 1.25e+81) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	} else if (B <= 1.9e+283) {
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -4e+146) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.02e-55) {
		tmp = Math.sqrt((2.0 * ((C + (A + Math.hypot(B, (A - C)))) * (F * t_1)))) * (-1.0 / t_1);
	} else if (B <= 1.25e+81) {
		tmp = (Math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -Math.sqrt((2.0 * C))) / t_1;
	} else if (B <= 1.9e+283) {
		tmp = Math.sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -4e+146:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.02e-55:
		tmp = math.sqrt((2.0 * ((C + (A + math.hypot(B, (A - C)))) * (F * t_1)))) * (-1.0 / t_1)
	elif B <= 1.25e+81:
		tmp = (math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -math.sqrt((2.0 * C))) / t_1
	elif B <= 1.9e+283:
		tmp = math.sqrt(((C * F) + (B * F))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -4e+146)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.02e-55)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(C + Float64(A + hypot(B, Float64(A - C)))) * Float64(F * t_1)))) * Float64(-1.0 / t_1));
	elseif (B <= 1.25e+81)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif (B <= 1.9e+283)
		tmp = Float64(sqrt(Float64(Float64(C * F) + Float64(B * F))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -4e+146)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.02e-55)
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * (F * t_1)))) * (-1.0 / t_1);
	elseif (B <= 1.25e+81)
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	elseif (B <= 1.9e+283)
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4e+146], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.02e-55], N[(N[Sqrt[N[(2.0 * N[(N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.25e+81], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 1.9e+283], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] + N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -3.99999999999999973e146

   1. Initial program 0.1%

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

      1. associate-*l* 0.1%

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

      2. unpow2 0.1%

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

      3. +-commutative 0.1%

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

      4. unpow2 0.1%

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

      5. associate-*l* 0.1%

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

      6. unpow2 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in A around -inf 2.4%

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

   5. Taylor expanded in B around -inf 8.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.1%

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

   7. Simplified 8.1%

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

   if -3.99999999999999973e146 < B < -1.02e-55

   1. Initial program 38.5%

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

      1. associate-*l* 38.5%

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

      2. unpow2 38.5%

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

      3. +-commutative 38.5%

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

      4. unpow2 38.5%

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

      5. associate-*l* 38.5%

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

      6. unpow2 38.5%

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

   3. Simplified 38.5%

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

      1. div-inv 38.5%

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

   5. Applied egg-rr 43.3%

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

   if -1.02e-55 < B < 1.25e81

   1. Initial program 21.9%

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

      1. associate-*l* 21.8%

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

      2. unpow2 21.8%

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

      3. +-commutative 21.8%

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

      4. unpow2 21.8%

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

      5. associate-*l* 21.8%

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

      6. unpow2 21.8%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in A around -inf 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 19.7%

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

      2. *-commutative 19.7%

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

      3. *-commutative 19.7%

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

   6. Applied egg-rr 19.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 19.7%

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

      2. unpow2 19.7%

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

      3. cancel-sign-sub-inv 19.7%

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

      4. unpow2 19.7%

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

      5. metadata-eval 19.7%

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

   8. Simplified 19.7%

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

   if 1.25e81 < B < 1.9000000000000001e283

   1. Initial program 9.7%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

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

      2. *-commutative 16.4%

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

      3. unpow2 16.4%

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

      4. unpow2 16.4%

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

   6. Simplified 16.4%

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

   7. Taylor expanded in C around 0 61.4%

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

   if 1.9000000000000001e283 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 0.0%

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

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

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

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{+146}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+283}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10: 36.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{+130}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -1.6e+130)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.6e-40)
       (- (/ (sqrt (* (* 2.0 (* F t_1)) (- A (- B C)))) t_1))
       (if (<= B 5.4e+80)
         (/
          (*
           (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
           (- (sqrt (* 2.0 C))))
          t_1)
         (if (<= B 3e+280)
           (* (sqrt (+ (* C F) (* B F))) (/ t_0 B))
           (* (sqrt (/ F B)) t_0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.6e+130) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.6e-40) {
		tmp = -(sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	} else if (B <= 5.4e+80) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	} else if (B <= 3e+280) {
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    t_1 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-1.6d+130)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-1.6d-40)) then
        tmp = -(sqrt(((2.0d0 * (f * t_1)) * (a - (b - c)))) / t_1)
    else if (b <= 5.4d+80) then
        tmp = (sqrt(((2.0d0 * f) * ((b * b) + ((-4.0d0) * (a * c))))) * -sqrt((2.0d0 * c))) / t_1
    else if (b <= 3d+280) then
        tmp = sqrt(((c * f) + (b * f))) * (t_0 / b)
    else
        tmp = sqrt((f / b)) * t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -1.6e+130) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.6e-40) {
		tmp = -(Math.sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	} else if (B <= 5.4e+80) {
		tmp = (Math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -Math.sqrt((2.0 * C))) / t_1;
	} else if (B <= 3e+280) {
		tmp = Math.sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -1.6e+130:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.6e-40:
		tmp = -(math.sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1)
	elif B <= 5.4e+80:
		tmp = (math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -math.sqrt((2.0 * C))) / t_1
	elif B <= 3e+280:
		tmp = math.sqrt(((C * F) + (B * F))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -1.6e+130)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.6e-40)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A - Float64(B - C)))) / t_1));
	elseif (B <= 5.4e+80)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif (B <= 3e+280)
		tmp = Float64(sqrt(Float64(Float64(C * F) + Float64(B * F))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -1.6e+130)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.6e-40)
		tmp = -(sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	elseif (B <= 5.4e+80)
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	elseif (B <= 3e+280)
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.6e+130], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.6e-40], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A - N[(B - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[B, 5.4e+80], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 3e+280], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] + N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -1.6e130

   1. Initial program 2.4%

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

      1. associate-*l* 2.4%

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

      2. unpow2 2.4%

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

      3. +-commutative 2.4%

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

      4. unpow2 2.4%

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

      5. associate-*l* 2.4%

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

      6. unpow2 2.4%

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

   3. Simplified 2.4%

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

   4. Taylor expanded in A around -inf 3.0%

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

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

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

   7. Simplified 8.6%

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

   if -1.6e130 < B < -1.60000000000000001e-40

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. unpow2 38.9%

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

      3. +-commutative 38.9%

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

      4. unpow2 38.9%

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

      5. associate-*l* 38.9%

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

      6. unpow2 38.9%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in B around -inf 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C + -1 \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 39.3%

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

      2. unsub-neg 39.3%

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

   6. Simplified 39.3%

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

   if -1.60000000000000001e-40 < B < 5.39999999999999966e80

   1. Initial program 21.9%

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

      1. associate-*l* 21.8%

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

      2. unpow2 21.8%

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

      3. +-commutative 21.8%

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

      4. unpow2 21.8%

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

      5. associate-*l* 21.8%

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

      6. unpow2 21.8%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in A around -inf 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 19.7%

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

      2. *-commutative 19.7%

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

      3. *-commutative 19.7%

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

   6. Applied egg-rr 19.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 19.7%

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

      2. unpow2 19.7%

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

      3. cancel-sign-sub-inv 19.7%

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

      4. unpow2 19.7%

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

      5. metadata-eval 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Simplified 19.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 5.39999999999999966e80 < B < 3.0000000000000001e280

   1. Initial program 9.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      2. *-commutative 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \]

      4. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \]

   6. Simplified 16.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]

   7. Taylor expanded in C around 0 61.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot B + C \cdot F}} \]

   if 3.0000000000000001e280 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in C around 0 0.0%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.6 \cdot 10^{+130}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11: 38.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -9.8 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+279}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -9.8e+148)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.7e-48)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ C (hypot B C))))) t_1)
       (if (<= B 5.2e+82)
         (/
          (*
           (sqrt (* (* 2.0 F) (+ (* B B) (* -4.0 (* A C)))))
           (- (sqrt (* 2.0 C))))
          t_1)
         (if (<= B 5.2e+279)
           (* (sqrt (+ (* C F) (* B F))) (/ t_0 B))
           (* (sqrt (/ F B)) t_0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -9.8e+148) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-48) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (C + hypot(B, C)))) / t_1;
	} else if (B <= 5.2e+82) {
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	} else if (B <= 5.2e+279) {
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -9.8e+148) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-48) {
		tmp = -Math.sqrt(((2.0 * (F * t_1)) * (C + Math.hypot(B, C)))) / t_1;
	} else if (B <= 5.2e+82) {
		tmp = (Math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -Math.sqrt((2.0 * C))) / t_1;
	} else if (B <= 5.2e+279) {
		tmp = Math.sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -9.8e+148:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.7e-48:
		tmp = -math.sqrt(((2.0 * (F * t_1)) * (C + math.hypot(B, C)))) / t_1
	elif B <= 5.2e+82:
		tmp = (math.sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -math.sqrt((2.0 * C))) / t_1
	elif B <= 5.2e+279:
		tmp = math.sqrt(((C * F) + (B * F))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -9.8e+148)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.7e-48)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(C + hypot(B, C))))) / t_1);
	elseif (B <= 5.2e+82)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_1);
	elseif (B <= 5.2e+279)
		tmp = Float64(sqrt(Float64(Float64(C * F) + Float64(B * F))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -9.8e+148)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.7e-48)
		tmp = -sqrt(((2.0 * (F * t_1)) * (C + hypot(B, C)))) / t_1;
	elseif (B <= 5.2e+82)
		tmp = (sqrt(((2.0 * F) * ((B * B) + (-4.0 * (A * C))))) * -sqrt((2.0 * C))) / t_1;
	elseif (B <= 5.2e+279)
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -9.8e+148], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.7e-48], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 5.2e+82], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 5.2e+279], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] + N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -9.8e148

   1. Initial program 0.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 0.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 0.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 5.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.9%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 5.9%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -9.8e148 < B < -1.70000000000000014e-48

   1. Initial program 37.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 37.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around 0 36.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. unpow2 36.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 36.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. hypot-def 39.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \color{blue}{\mathsf{hypot}\left(B, C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 39.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(C + \mathsf{hypot}\left(B, C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.70000000000000014e-48 < B < 5.1999999999999997e82

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. sqrt-prod 19.7%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 19.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right) \cdot F\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 19.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Applied egg-rr 19.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 19.7%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)}} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\color{blue}{{B}^{2}} - 4 \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(C \cdot A\right)\right)}} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Simplified 19.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)} \cdot \sqrt{2 \cdot C}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 5.1999999999999997e82 < B < 5.2000000000000003e279

   1. Initial program 9.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      2. *-commutative 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \]

      4. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \]

   6. Simplified 16.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]

   7. Taylor expanded in C around 0 61.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot B + C \cdot F}} \]

   if 5.2000000000000003e279 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in C around 0 0.0%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.8 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+279}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12: 37.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-44}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+80}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 10^{+283}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -8.5e+131)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.15e-44)
       (- (/ (sqrt (* (* 2.0 (* F t_1)) (- A (- B C)))) t_1))
       (if (<= B 4e+80)
         (- (/ (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C)))))) t_1))
         (if (<= B 1e+283)
           (* (sqrt (+ (* C F) (* B F))) (/ t_0 B))
           (* (sqrt (/ F B)) t_0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -8.5e+131) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.15e-44) {
		tmp = -(sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	} else if (B <= 4e+80) {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	} else if (B <= 1e+283) {
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    t_1 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-8.5d+131)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-1.15d-44)) then
        tmp = -(sqrt(((2.0d0 * (f * t_1)) * (a - (b - c)))) / t_1)
    else if (b <= 4d+80) then
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / t_1)
    else if (b <= 1d+283) then
        tmp = sqrt(((c * f) + (b * f))) * (t_0 / b)
    else
        tmp = sqrt((f / b)) * t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -8.5e+131) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.15e-44) {
		tmp = -(Math.sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	} else if (B <= 4e+80) {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	} else if (B <= 1e+283) {
		tmp = Math.sqrt(((C * F) + (B * F))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -8.5e+131:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.15e-44:
		tmp = -(math.sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1)
	elif B <= 4e+80:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1)
	elif B <= 1e+283:
		tmp = math.sqrt(((C * F) + (B * F))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -8.5e+131)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.15e-44)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A - Float64(B - C)))) / t_1));
	elseif (B <= 4e+80)
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / t_1));
	elseif (B <= 1e+283)
		tmp = Float64(sqrt(Float64(Float64(C * F) + Float64(B * F))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	t_1 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -8.5e+131)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.15e-44)
		tmp = -(sqrt(((2.0 * (F * t_1)) * (A - (B - C)))) / t_1);
	elseif (B <= 4e+80)
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	elseif (B <= 1e+283)
		tmp = sqrt(((C * F) + (B * F))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -8.5e+131], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.15e-44], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A - N[(B - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[B, 4e+80], (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), If[LessEqual[B, 1e+283], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] + N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -8.50000000000000063e131

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -8.50000000000000063e131 < B < -1.14999999999999999e-44

   1. Initial program 38.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in B around -inf 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C + -1 \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(C - B\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C - B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.14999999999999999e-44 < B < 4e80

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 16.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 16.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 16.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 4e80 < B < 9.99999999999999955e282

   1. Initial program 9.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      2. *-commutative 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \]

      4. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \]

   6. Simplified 16.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]

   7. Taylor expanded in C around 0 61.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot B + C \cdot F}} \]

   if 9.99999999999999955e282 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in C around 0 0.0%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.5 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-44}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+80}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 10^{+283}:\\ \;\;\;\;\sqrt{C \cdot F + B \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13: 36.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -6.4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-42}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+82}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -6.4e+131)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1e-42)
       (- (/ (sqrt (* (* 2.0 (* F t_0)) (- A (- B C)))) t_0))
       (if (<= B 4.8e+82)
         (- (/ (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C)))))) t_0))
         (if (<= B 3.8e+280)
           (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
           (* (sqrt (/ F B)) (- (sqrt 2.0)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -6.4e+131) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1e-42) {
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else if (B <= 4.8e+82) {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	} else if (B <= 3.8e+280) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-6.4d+131)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-1d-42)) then
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * (a - (b - c)))) / t_0)
    else if (b <= 4.8d+82) then
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / t_0)
    else if (b <= 3.8d+280) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * f))
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -6.4e+131) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1e-42) {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else if (B <= 4.8e+82) {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	} else if (B <= 3.8e+280) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -6.4e+131:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1e-42:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0)
	elif B <= 4.8e+82:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0)
	elif B <= 3.8e+280:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -6.4e+131)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1e-42)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A - Float64(B - C)))) / t_0));
	elseif (B <= 4.8e+82)
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / t_0));
	elseif (B <= 3.8e+280)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -6.4e+131)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1e-42)
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	elseif (B <= 4.8e+82)
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	elseif (B <= 3.8e+280)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -6.4e+131], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1e-42], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A - N[(B - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 4.8e+82], (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 3.8e+280], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -6.4000000000000004e131

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -6.4000000000000004e131 < B < -1.00000000000000004e-42

   1. Initial program 38.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in B around -inf 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C + -1 \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(C - B\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C - B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.00000000000000004e-42 < B < 4.79999999999999996e82

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 21.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 16.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 16.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 16.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 16.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 16.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 4.79999999999999996e82 < B < 3.79999999999999964e280

   1. Initial program 9.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 10.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in A around 0 16.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 16.4%

         \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      2. *-commutative 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)} \]

      4. unpow2 16.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)} \]

   6. Simplified 16.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}} \]

   7. Taylor expanded in C around 0 61.4%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]

   if 3.79999999999999964e280 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in C around 0 0.0%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 64.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   7. Simplified 64.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6.4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-42}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{+82}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{+280}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 14: 37.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -9.6 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-49}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -9.6e+131)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -7.5e-49)
       (- (/ (sqrt (* (* 2.0 (* F t_0)) (- A (- B C)))) t_0))
       (if (<= B 4.2e+62)
         (- (/ (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C)))))) t_0))
         (* (sqrt (/ F B)) (- (sqrt 2.0))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -9.6e+131) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -7.5e-49) {
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else if (B <= 4.2e+62) {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-9.6d+131)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-7.5d-49)) then
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * (a - (b - c)))) / t_0)
    else if (b <= 4.2d+62) then
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / t_0)
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -9.6e+131) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -7.5e-49) {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else if (B <= 4.2e+62) {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -9.6e+131:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -7.5e-49:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0)
	elif B <= 4.2e+62:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0)
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -9.6e+131)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -7.5e-49)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A - Float64(B - C)))) / t_0));
	elseif (B <= 4.2e+62)
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / t_0));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -9.6e+131)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -7.5e-49)
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	elseif (B <= 4.2e+62)
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -9.6e+131], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7.5e-49], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A - N[(B - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[B, 4.2e+62], (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -9.5999999999999998e131

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -9.5999999999999998e131 < B < -7.4999999999999998e-49

   1. Initial program 38.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in B around -inf 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C + -1 \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(C - B\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C - B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -7.4999999999999998e-49 < B < 4.2e62

   1. Initial program 22.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 22.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 22.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 16.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 16.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 16.5%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 16.5%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 16.5%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 16.5%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 16.5%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 16.5%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 16.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 4.2e62 < B

   1. Initial program 7.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]

   4. Taylor expanded in C around 0 10.9%

      \[\leadsto \frac{-\color{blue}{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 50.1%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 50.1%

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   7. Simplified 50.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.6 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-49}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+62}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 15: 28.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -4.5e+130)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -2.6e-40)
       (- (/ (sqrt (* (* 2.0 (* F t_0)) (- A (- B C)))) t_0))
       (- (/ (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C)))))) t_0))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -4.5e+130) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -2.6e-40) {
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-4.5d+130)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-2.6d-40)) then
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * (a - (b - c)))) / t_0)
    else
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / t_0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -4.5e+130) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -2.6e-40) {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	} else {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -4.5e+130:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -2.6e-40:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0)
	else:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -4.5e+130)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -2.6e-40)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A - Float64(B - C)))) / t_0));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / t_0));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -4.5e+130)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -2.6e-40)
		tmp = -(sqrt(((2.0 * (F * t_0)) * (A - (B - C)))) / t_0);
	else
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4.5e+130], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.6e-40], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A - N[(B - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < -4.50000000000000039e130

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -4.50000000000000039e130 < B < -2.6000000000000001e-40

   1. Initial program 38.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in B around -inf 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C + -1 \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \color{blue}{\left(-B\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 39.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(C - B\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 39.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(C - B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -2.6000000000000001e-40 < B

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 11.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 11.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 11.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 11.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A - \left(B - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 16: 20.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B -5.5e+14)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B 2.6e-71)
       (- (/ (sqrt (* (* A -16.0) (* F (* C C)))) t_0))
       (/ (- (sqrt (* B (* 2.0 (* F t_0))))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -5.5e+14) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= 2.6e-71) {
		tmp = -(sqrt(((A * -16.0) * (F * (C * C)))) / t_0);
	} else {
		tmp = -sqrt((B * (2.0 * (F * t_0)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= (-5.5d+14)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= 2.6d-71) then
        tmp = -(sqrt(((a * (-16.0d0)) * (f * (c * c)))) / t_0)
    else
        tmp = -sqrt((b * (2.0d0 * (f * t_0)))) / t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= -5.5e+14) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= 2.6e-71) {
		tmp = -(Math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0);
	} else {
		tmp = -Math.sqrt((B * (2.0 * (F * t_0)))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= -5.5e+14:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= 2.6e-71:
		tmp = -(math.sqrt(((A * -16.0) * (F * (C * C)))) / t_0)
	else:
		tmp = -math.sqrt((B * (2.0 * (F * t_0)))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= -5.5e+14)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= 2.6e-71)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))) / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(B * Float64(2.0 * Float64(F * t_0))))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= -5.5e+14)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= 2.6e-71)
		tmp = -(sqrt(((A * -16.0) * (F * (C * C)))) / t_0);
	else
		tmp = -sqrt((B * (2.0 * (F * t_0)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -5.5e+14], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.6e-71], (-N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[((-N[Sqrt[N[(B * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -5.5e14

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 18.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 2.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 6.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 6.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -5.5e14 < B < 2.5999999999999999e-71

   1. Initial program 23.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 23.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 23.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 16.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around 0 10.9%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 10.9%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 10.9%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(F \cdot {C}^{2}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. unpow2 10.9%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \color{blue}{\left(C \cdot C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 10.9%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 2.5999999999999999e-71 < B

   1. Initial program 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 11.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. unpow2 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. hypot-udef 15.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-un-lft-identity 15.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 15.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 15.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 15.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around inf 11.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{B}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 17: 28.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - t_0\\ \mathbf{if}\;B \leq -4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))) (t_1 (- (* B B) t_0)))
   (if (<= B -4e+131)
     (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
     (if (<= B -1.7e-40)
       (/ (- (sqrt (* B (* 2.0 (* F (- t_0 (* B B))))))) t_1)
       (- (/ (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C)))))) t_1))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = (B * B) - t_0;
	double tmp;
	if (B <= -4e+131) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-40) {
		tmp = -sqrt((B * (2.0 * (F * (t_0 - (B * B)))))) / t_1;
	} else {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (a * c)
    t_1 = (b * b) - t_0
    if (b <= (-4d+131)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else if (b <= (-1.7d-40)) then
        tmp = -sqrt((b * (2.0d0 * (f * (t_0 - (b * b)))))) / t_1
    else
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / t_1)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double t_1 = (B * B) - t_0;
	double tmp;
	if (B <= -4e+131) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else if (B <= -1.7e-40) {
		tmp = -Math.sqrt((B * (2.0 * (F * (t_0 - (B * B)))))) / t_1;
	} else {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = 4.0 * (A * C)
	t_1 = (B * B) - t_0
	tmp = 0
	if B <= -4e+131:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	elif B <= -1.7e-40:
		tmp = -math.sqrt((B * (2.0 * (F * (t_0 - (B * B)))))) / t_1
	else:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	t_1 = Float64(Float64(B * B) - t_0)
	tmp = 0.0
	if (B <= -4e+131)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	elseif (B <= -1.7e-40)
		tmp = Float64(Float64(-sqrt(Float64(B * Float64(2.0 * Float64(F * Float64(t_0 - Float64(B * B))))))) / t_1);
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / t_1));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = 4.0 * (A * C);
	t_1 = (B * B) - t_0;
	tmp = 0.0;
	if (B <= -4e+131)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	elseif (B <= -1.7e-40)
		tmp = -sqrt((B * (2.0 * (F * (t_0 - (B * B)))))) / t_1;
	else
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[B, -4e+131], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.7e-40], N[((-N[Sqrt[N[(B * N[(2.0 * N[(F * N[(t$95$0 - N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.9999999999999996e131

   1. Initial program 2.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 2.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 8.6%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 8.6%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -3.9999999999999996e131 < B < -1.69999999999999992e-40

   1. Initial program 38.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. unpow2 38.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. hypot-udef 44.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-un-lft-identity 44.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 44.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{1 \cdot \mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 44.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 44.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around -inf 38.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(-1 \cdot B\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 38.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(-B\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 38.4%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(-B\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.69999999999999992e-40 < B

   1. Initial program 17.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 11.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 11.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 11.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 11.4%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 11.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{+131}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 18: 28.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -6e+72)
   (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
   (-
    (/
     (sqrt (* (* 4.0 C) (* F (+ (* B B) (* -4.0 (* A C))))))
     (- (* B B) (* 4.0 (* A C)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+72) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-6d+72)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else
        tmp = -(sqrt(((4.0d0 * c) * (f * ((b * b) + ((-4.0d0) * (a * c)))))) / ((b * b) - (4.0d0 * (a * c))))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -6e+72) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -(Math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -6e+72:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	else:
		tmp = -(math.sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -6e+72)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -6e+72)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	else
		tmp = -(sqrt(((4.0 * C) * (F * ((B * B) + (-4.0 * (A * C)))))) / ((B * B) - (4.0 * (A * C))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -6e+72], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < -6.00000000000000006e72

   1. Initial program 12.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 12.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 12.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 3.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 7.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 7.5%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 7.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -6.00000000000000006e72 < B

   1. Initial program 19.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 19.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 19.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 10.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 10.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{4 \cdot \left(C \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 10.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 10.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \color{blue}{\left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. cancel-sign-sub-inv 10.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \color{blue}{\left({B}^{2} + \left(-4\right) \cdot \left(A \cdot C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. unpow2 10.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + \left(-4\right) \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. metadata-eval 10.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. *-commutative 10.2%

         \[\leadsto \frac{-\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 10.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{+72}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 19: 18.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -1.4 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -1.4e+17)
   (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
   (- (/ (sqrt (* (* A -16.0) (* F (* C C)))) (- (* B B) (* 4.0 (* A C)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.4e+17) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -(sqrt(((A * -16.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C))));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-1.4d+17)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else
        tmp = -(sqrt(((a * (-16.0d0)) * (f * (c * c)))) / ((b * b) - (4.0d0 * (a * c))))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.4e+17) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -(Math.sqrt(((A * -16.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C))));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -1.4e+17:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	else:
		tmp = -(math.sqrt(((A * -16.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C))))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -1.4e+17)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * Float64(C * C)))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -1.4e+17)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	else
		tmp = -(sqrt(((A * -16.0) * (F * (C * C)))) / ((B * B) - (4.0 * (A * C))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -1.4e+17], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < -1.4e17

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 18.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 2.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 6.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 6.5%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 6.5%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -1.4e17 < B

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 18.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 10.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around 0 7.6%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 7.6%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. *-commutative 7.6%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(F \cdot {C}^{2}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. unpow2 7.6%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(F \cdot \color{blue}{\left(C \cdot C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 7.6%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.4 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 20: 9.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{B} \cdot {\left(C \cdot F\right)}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -2e-310)
   (* 2.0 (* (sqrt (* C F)) (/ 1.0 B)))
   (* -2.0 (* (/ 1.0 B) (pow (* C F) 0.5)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -2.0 * ((1.0 / B) * pow((C * F), 0.5));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = 2.0d0 * (sqrt((c * f)) * (1.0d0 / b))
    else
        tmp = (-2.0d0) * ((1.0d0 / b) * ((c * f) ** 0.5d0))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (Math.sqrt((C * F)) * (1.0 / B));
	} else {
		tmp = -2.0 * ((1.0 / B) * Math.pow((C * F), 0.5));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -2e-310:
		tmp = 2.0 * (math.sqrt((C * F)) * (1.0 / B))
	else:
		tmp = -2.0 * ((1.0 / B) * math.pow((C * F), 0.5))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -2e-310)
		tmp = Float64(2.0 * Float64(sqrt(Float64(C * F)) * Float64(1.0 / B)));
	else
		tmp = Float64(-2.0 * Float64(Float64(1.0 / B) * (Float64(C * F) ^ 0.5)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -2e-310)
		tmp = 2.0 * (sqrt((C * F)) * (1.0 / B));
	else
		tmp = -2.0 * ((1.0 / B) * ((C * F) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -2e-310], N[(2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(1.0 / B), $MachinePrecision] * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -1.999999999999994e-310

   1. Initial program 23.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 23.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 7.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 5.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.1%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 5.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -1.999999999999994e-310 < B

   1. Initial program 13.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 13.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 9.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 2.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 2.6%

         \[\leadsto -2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 2.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]
   8. Step-by-step derivation

      1. pow1/2 2.8%

         \[\leadsto -2 \cdot \left(\color{blue}{{\left(F \cdot C\right)}^{0.5}} \cdot \frac{1}{B}\right) \]

   9. Applied egg-rr 2.8%

      \[\leadsto -2 \cdot \left(\color{blue}{{\left(F \cdot C\right)}^{0.5}} \cdot \frac{1}{B}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{B} \cdot {\left(C \cdot F\right)}^{0.5}\right)\\ \end{array} \]

Alternative 21: 9.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{C \cdot F}\\ \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* C F))))
   (if (<= B -2e-310) (* 2.0 (* t_0 (/ 1.0 B))) (* -2.0 (/ t_0 B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((C * F));
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((c * f))
    if (b <= (-2d-310)) then
        tmp = 2.0d0 * (t_0 * (1.0d0 / b))
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((C * F));
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * (t_0 * (1.0 / B));
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((C * F))
	tmp = 0
	if B <= -2e-310:
		tmp = 2.0 * (t_0 * (1.0 / B))
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(C * F))
	tmp = 0.0
	if (B <= -2e-310)
		tmp = Float64(2.0 * Float64(t_0 * Float64(1.0 / B)));
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((C * F));
	tmp = 0.0;
	if (B <= -2e-310)
		tmp = 2.0 * (t_0 * (1.0 / B));
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -2e-310], N[(2.0 * N[(t$95$0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -1.999999999999994e-310

   1. Initial program 23.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 23.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 7.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 5.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.1%

         \[\leadsto 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 5.1%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   if -1.999999999999994e-310 < B

   1. Initial program 13.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 13.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 13.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 9.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 2.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 2.6%

         \[\leadsto -2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   7. Simplified 2.6%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

   8. Taylor expanded in B around 0 2.6%

      \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 2.6%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-commutative 2.6%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}} \cdot 1}{B} \]

      3. *-rgt-identity 2.6%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot C}}}{B} \]

   10. Simplified 2.6%

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 3.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 22: 5.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ -2 \cdot \frac{\sqrt{C \cdot F}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* C F)) B)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((C * F)) / B);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (sqrt((c * f)) / b)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((C * F)) / B);
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((C * F)) / B)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((C * F)) / B);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. associate-*l* 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. unpow2 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. +-commutative 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. unpow2 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. associate-*l* 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   6. unpow2 18.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

3. Simplified 18.2%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

4. Taylor expanded in A around -inf 8.6%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

5. Taylor expanded in B around inf 2.1%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
6. Step-by-step derivation

   1. *-commutative 2.1%

      \[\leadsto -2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

7. Simplified 2.1%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)} \]

8. Taylor expanded in B around 0 2.1%

   \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 2.1%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

   2. *-commutative 2.1%

      \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}} \cdot 1}{B} \]

   3. *-rgt-identity 2.1%

      \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot C}}}{B} \]

10. Simplified 2.1%

    \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{F \cdot C}}{B}} \]

11. Final simplification 2.1%

    \[\leadsto -2 \cdot \frac{\sqrt{C \cdot F}}{B} \]

Reproduce

herbie shell --seed 2023181 
(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))))