ABCF->ab-angle a

Percentage Accurate: 19.0% → 52.1%

Time: 31.3s

Alternatives: 20

Speedup: 4.8×

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: 19.0% 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: 52.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= (pow B 2.0) 5e+203)
     (/
      (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A (hypot B (- A C)))))))
      t_0)
     (* (/ (sqrt 2.0) B) (* (sqrt (+ C (hypot B C))) (- (sqrt F)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (pow(B, 2.0) <= 5e+203) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(B, C))) * -sqrt(F));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (Math.pow(B, 2.0) <= 5e+203) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (A + Math.hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt((C + Math.hypot(B, C))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if math.pow(B, 2.0) <= 5e+203:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (A + math.hypot(B, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt((C + math.hypot(B, C))) * -math.sqrt(F))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e+203)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(B, C))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if ((B ^ 2.0) <= 5e+203)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(B, C))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+203], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $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[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 4.99999999999999994e203

   1. Initial program 24.2%

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

      1. associate-*l* 24.2%

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

      2. unpow2 24.2%

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

      3. +-commutative 24.2%

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

      4. unpow2 24.2%

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

      5. associate-*l* 24.2%

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

      6. unpow2 24.2%

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

   3. Simplified 24.2%

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

      1. sqrt-prod 26.3%

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

      2. *-commutative 26.3%

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

      3. *-commutative 26.3%

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

      4. associate-+l+ 26.9%

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

      5. unpow2 26.9%

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

      6. hypot-udef 41.6%

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

      7. associate-+r+ 40.5%

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

      8. +-commutative 40.5%

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

      9. associate-+r+ 41.3%

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

   5. Applied egg-rr 41.3%

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

   if 4.99999999999999994e203 < (pow.f64 B 2)

   1. Initial program 3.8%

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

   3. Simplified 3.9%

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

   4. Taylor expanded in A around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. distribute-rgt-neg-in 6.1%

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

      3. *-commutative 6.1%

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

      4. unpow2 6.1%

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

      5. unpow2 6.1%

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

      6. hypot-def 25.1%

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

   6. Simplified 25.1%

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

      1. sqrt-prod 34.5%

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

   8. Applied egg-rr 34.5%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2: 50.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 2.25 \cdot 10^{+111}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B 2.25e+111)
     (/
      (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A (hypot B (- A C)))))))
      t_0)
     (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt (+ B C))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 2.25e+111) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt((B + C)));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 2.25e+111) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (A + Math.hypot(B, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(F) * -Math.sqrt((B + C)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= 2.25e+111:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (A + math.hypot(B, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(F) * -math.sqrt((B + C)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= 2.25e+111)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(Float64(B + C)))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= 2.25e+111)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + hypot(B, (A - C)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt((B + C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 2.25e+111], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $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[N[(B + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.25e111

   1. Initial program 19.4%

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

      1. associate-*l* 19.4%

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

      2. unpow2 19.4%

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

      3. +-commutative 19.4%

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

      4. unpow2 19.4%

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

      5. associate-*l* 19.4%

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

      6. unpow2 19.4%

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

   3. Simplified 19.4%

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

      1. sqrt-prod 22.8%

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

      2. *-commutative 22.8%

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

      3. *-commutative 22.8%

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

      4. associate-+l+ 23.3%

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

      5. unpow2 23.3%

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

      6. hypot-udef 34.4%

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

      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+ 34.1%

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

   5. Applied egg-rr 34.1%

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

   if 2.25e111 < B

   1. Initial program 3.1%

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

   3. Simplified 3.2%

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

   4. Taylor expanded in A around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. distribute-rgt-neg-in 9.8%

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

      3. *-commutative 9.8%

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

      4. unpow2 9.8%

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

      5. unpow2 9.8%

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

      6. hypot-def 50.8%

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

   6. Simplified 50.8%

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

      1. sqrt-prod 72.0%

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

   8. Applied egg-rr 72.0%

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

   9. Taylor expanded in C around 0 66.6%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.25 \cdot 10^{+111}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]

Alternative 3: 46.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 6.6 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B 6.6e-242)
     (/ (- (sqrt (* 2.0 (* 2.0 (* (* F C) (fma B B (* C (* A -4.0)))))))) t_0)
     (if (<= B 6.2e-125)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (* 2.0 A))))) t_0)
       (if (<= B 1.55e+88)
         (- (/ (sqrt (* 2.0 (* (* F t_1) (+ C (+ A (hypot B (- A C))))))) t_1))
         (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 6.6e-242) {
		tmp = -sqrt((2.0 * (2.0 * ((F * C) * fma(B, B, (C * (A * -4.0))))))) / t_0;
	} else if (B <= 6.2e-125) {
		tmp = -sqrt((2.0 * ((F * t_0) * (2.0 * A)))) / t_0;
	} else if (B <= 1.55e+88) {
		tmp = -(sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1);
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= 6.6e-242)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(2.0 * Float64(Float64(F * C) * fma(B, B, Float64(C * Float64(A * -4.0)))))))) / t_0);
	elseif (B <= 6.2e-125)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(2.0 * A))))) / t_0);
	elseif (B <= 1.55e+88)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 6.6e-242], N[((-N[Sqrt[N[(2.0 * N[(2.0 * N[(N[(F * C), $MachinePrecision] * N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 6.2e-125], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.55e+88], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 6.59999999999999963e-242

   1. Initial program 16.0%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 12.6%

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

   5. Taylor expanded in F around 0 12.0%

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

      1. metadata-eval 12.0%

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

      2. cancel-sign-sub-inv 12.0%

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

      3. *-commutative 12.0%

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

      4. *-commutative 12.0%

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

      5. cancel-sign-sub-inv 12.0%

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

      6. metadata-eval 12.0%

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

      7. associate-*r* 12.0%

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

      8. *-commutative 12.0%

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

      9. unpow2 12.0%

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

      10. fma-def 12.0%

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

      11. *-commutative 12.0%

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

   7. Simplified 12.0%

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

   if 6.59999999999999963e-242 < B < 6.20000000000000026e-125

   1. Initial program 8.6%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in C around -inf 26.9%

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

   if 6.20000000000000026e-125 < B < 1.5500000000000001e88

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. unpow2 36.1%

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

      3. +-commutative 36.1%

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

      4. unpow2 36.1%

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

      5. associate-*l* 36.1%

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

      6. unpow2 36.1%

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

   3. Simplified 36.1%

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

      1. distribute-frac-neg 36.1%

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

   5. Applied egg-rr 48.7%

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

   if 1.5500000000000001e88 < B

   1. Initial program 7.3%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in A around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. distribute-rgt-neg-in 17.0%

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

      3. *-commutative 17.0%

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

      4. unpow2 17.0%

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

      5. unpow2 17.0%

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

      6. hypot-def 53.6%

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

   6. Simplified 53.6%

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

      1. sqrt-prod 72.5%

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

   8. Applied egg-rr 72.5%

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

   9. Taylor expanded in C around 0 67.6%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.6 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 4: 45.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 9 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B 9e-242)
     (/
      (- (sqrt (* 2.0 (* 2.0 (* (* F C) (fma B B (* C (* A -4.0))))))))
      (fma C (* A -4.0) (* B B)))
     (if (<= B 1.75e-124)
       (/
        (- (sqrt (* 2.0 (* F (* (* 2.0 A) (+ (* B B) (* (* C A) -4.0)))))))
        t_0)
       (if (<= B 1.95e+88)
         (- (/ (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C))))))) t_0))
         (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 9e-242) {
		tmp = -sqrt((2.0 * (2.0 * ((F * C) * fma(B, B, (C * (A * -4.0))))))) / fma(C, (A * -4.0), (B * B));
	} else if (B <= 1.75e-124) {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * ((B * B) + ((C * A) * -4.0)))))) / t_0;
	} else if (B <= 1.95e+88) {
		tmp = -(sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0);
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= 9e-242)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(2.0 * Float64(Float64(F * C) * fma(B, B, Float64(C * Float64(A * -4.0)))))))) / fma(C, Float64(A * -4.0), Float64(B * B)));
	elseif (B <= 1.75e-124)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))))) / t_0);
	elseif (B <= 1.95e+88)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9e-242], N[((-N[Sqrt[N[(2.0 * N[(2.0 * N[(N[(F * C), $MachinePrecision] * N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.75e-124], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.95e+88], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.9999999999999997e-242

   1. Initial program 16.0%

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

   3. Simplified 20.1%

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

   4. Taylor expanded in A around -inf 12.6%

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

   5. Taylor expanded in F around 0 12.0%

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

      1. metadata-eval 12.0%

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

      2. cancel-sign-sub-inv 12.0%

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

      3. *-commutative 12.0%

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

      4. *-commutative 12.0%

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

      5. cancel-sign-sub-inv 12.0%

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

      6. metadata-eval 12.0%

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

      7. associate-*r* 12.0%

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

      8. *-commutative 12.0%

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

      9. unpow2 12.0%

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

      10. fma-def 12.0%

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

      11. *-commutative 12.0%

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

   7. Simplified 12.0%

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

   if 8.9999999999999997e-242 < B < 1.7499999999999999e-124

   1. Initial program 8.6%

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

      1. associate-*l* 8.6%

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

      2. unpow2 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. associate-*l* 8.6%

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

      6. unpow2 8.6%

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

   3. Simplified 8.6%

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

      1. *-un-lft-identity 8.6%

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

      2. associate-*l* 8.6%

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

      3. *-commutative 8.6%

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

      4. *-commutative 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-udef 12.2%

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

      7. +-commutative 12.2%

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

      8. associate-+r+ 13.3%

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

   5. Applied egg-rr 13.3%

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

      1. *-lft-identity 13.3%

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

      2. associate-*r* 13.3%

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

      3. fma-neg 13.3%

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

      4. *-commutative 13.3%

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

      5. *-commutative 13.3%

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

      6. distribute-rgt-neg-in 13.3%

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

      7. *-commutative 13.3%

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

      8. metadata-eval 13.3%

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

      9. associate-*r* 13.3%

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

      10. associate-*r* 13.3%

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

      11. associate-*l* 13.3%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in A around inf 5.5%

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

      1. associate-+r+ 23.8%

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

      2. distribute-rgt1-in 23.8%

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

      3. metadata-eval 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 23.8%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.7499999999999999e-124 < B < 1.9500000000000001e88

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. unpow2 36.1%

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

      3. +-commutative 36.1%

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

      4. unpow2 36.1%

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

      5. associate-*l* 36.1%

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

      6. unpow2 36.1%

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

   3. Simplified 36.1%

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

      1. distribute-frac-neg 36.1%

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

   5. Applied egg-rr 48.7%

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

   if 1.9500000000000001e88 < B

   1. Initial program 7.3%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in A around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. distribute-rgt-neg-in 17.0%

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

      3. *-commutative 17.0%

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

      4. unpow2 17.0%

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

      5. unpow2 17.0%

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

      6. hypot-def 53.6%

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

   6. Simplified 53.6%

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

      1. sqrt-prod 72.5%

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

   8. Applied egg-rr 72.5%

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

   9. Taylor expanded in C around 0 67.6%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot C\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5: 45.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 7.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot t_0\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{+89}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B 7.4e-242)
     (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)
     (if (<= B 5.8e-123)
       (/ (- (sqrt (* 2.0 (* F (* (* 2.0 A) t_0))))) t_1)
       (if (<= B 1.85e+89)
         (- (/ (sqrt (* 2.0 (* (* F t_1) (+ C (+ A (hypot B (- A C))))))) t_1))
         (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 7.4e-242) {
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (B <= 5.8e-123) {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 1.85e+89) {
		tmp = -(sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1);
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 7.4e-242) {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (B <= 5.8e-123) {
		tmp = -Math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 1.85e+89) {
		tmp = -(Math.sqrt((2.0 * ((F * t_1) * (C + (A + Math.hypot(B, (A - C))))))) / t_1);
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= 7.4e-242:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	elif B <= 5.8e-123:
		tmp = -math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1
	elif B <= 1.85e+89:
		tmp = -(math.sqrt((2.0 * ((F * t_1) * (C + (A + math.hypot(B, (A - C))))))) / t_1)
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= 7.4e-242)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1);
	elseif (B <= 5.8e-123)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * t_0))))) / t_1);
	elseif (B <= 1.85e+89)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= 7.4e-242)
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	elseif (B <= 5.8e-123)
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	elseif (B <= 1.85e+89)
		tmp = -(sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1);
	else
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 7.4e-242], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 5.8e-123], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.85e+89], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 7.39999999999999994e-242

   1. Initial program 16.0%

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

      1. associate-*l* 16.0%

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

      2. unpow2 16.0%

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

      3. +-commutative 16.0%

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

      4. unpow2 16.0%

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

      5. associate-*l* 16.0%

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

      6. unpow2 16.0%

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

   3. Simplified 16.0%

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

      1. *-un-lft-identity 16.0%

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

      2. associate-*l* 16.0%

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

      3. *-commutative 16.0%

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

      4. *-commutative 16.0%

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

      5. unpow2 16.0%

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

      6. hypot-udef 19.0%

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

      7. +-commutative 19.0%

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

      8. associate-+r+ 20.1%

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

   5. Applied egg-rr 20.1%

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

      1. *-lft-identity 20.1%

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

      2. associate-*r* 20.1%

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

      3. fma-neg 20.1%

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

      4. *-commutative 20.1%

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

      5. *-commutative 20.1%

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

      6. distribute-rgt-neg-in 20.1%

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

      7. *-commutative 20.1%

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

      8. metadata-eval 20.1%

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

      9. associate-*r* 20.1%

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

      10. associate-*r* 20.1%

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

      11. associate-*l* 17.4%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in A around -inf 10.6%

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

      1. *-commutative 10.6%

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

   10. Simplified 10.6%

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

   if 7.39999999999999994e-242 < B < 5.80000000000000007e-123

   1. Initial program 8.6%

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

      1. associate-*l* 8.6%

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

      2. unpow2 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. associate-*l* 8.6%

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

      6. unpow2 8.6%

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

   3. Simplified 8.6%

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

      1. *-un-lft-identity 8.6%

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

      2. associate-*l* 8.6%

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

      3. *-commutative 8.6%

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

      4. *-commutative 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-udef 12.2%

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

      7. +-commutative 12.2%

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

      8. associate-+r+ 13.3%

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

   5. Applied egg-rr 13.3%

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

      1. *-lft-identity 13.3%

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

      2. associate-*r* 13.3%

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

      3. fma-neg 13.3%

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

      4. *-commutative 13.3%

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

      5. *-commutative 13.3%

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

      6. distribute-rgt-neg-in 13.3%

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

      7. *-commutative 13.3%

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

      8. metadata-eval 13.3%

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

      9. associate-*r* 13.3%

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

      10. associate-*r* 13.3%

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

      11. associate-*l* 13.3%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in A around inf 5.5%

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

      1. associate-+r+ 23.8%

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

      2. distribute-rgt1-in 23.8%

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

      3. metadata-eval 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 23.8%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 5.80000000000000007e-123 < B < 1.8499999999999999e89

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. unpow2 36.1%

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

      3. +-commutative 36.1%

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

      4. unpow2 36.1%

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

      5. associate-*l* 36.1%

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

      6. unpow2 36.1%

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

   3. Simplified 36.1%

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

      1. distribute-frac-neg 36.1%

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

   5. Applied egg-rr 48.7%

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

   if 1.8499999999999999e89 < B

   1. Initial program 7.3%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in A around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. distribute-rgt-neg-in 17.0%

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

      3. *-commutative 17.0%

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

      4. unpow2 17.0%

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

      5. unpow2 17.0%

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

      6. hypot-def 53.6%

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

   6. Simplified 53.6%

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

      1. sqrt-prod 72.5%

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

   8. Applied egg-rr 72.5%

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

   9. Taylor expanded in C around 0 67.6%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{+89}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 6: 36.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 9 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot t_0\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B 9e-242)
     (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)
     (if (<= B 1e-123)
       (/ (- (sqrt (* 2.0 (* F (* (* 2.0 A) t_0))))) t_1)
       (if (<= B 1.12e+88)
         (- (/ (sqrt (* 2.0 (* (* F t_1) (+ C (+ A (hypot B (- A C))))))) t_1))
         (* (/ (sqrt 2.0) B) (- (sqrt (* B F)))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 9e-242) {
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (B <= 1e-123) {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 1.12e+88) {
		tmp = -(sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1);
	} else {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= 9e-242) {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (B <= 1e-123) {
		tmp = -Math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 1.12e+88) {
		tmp = -(Math.sqrt((2.0 * ((F * t_1) * (C + (A + Math.hypot(B, (A - C))))))) / t_1);
	} else {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= 9e-242:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	elif B <= 1e-123:
		tmp = -math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1
	elif B <= 1.12e+88:
		tmp = -(math.sqrt((2.0 * ((F * t_1) * (C + (A + math.hypot(B, (A - C))))))) / t_1)
	else:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= 9e-242)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1);
	elseif (B <= 1e-123)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * t_0))))) / t_1);
	elseif (B <= 1.12e+88)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) / t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= 9e-242)
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	elseif (B <= 1e-123)
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	elseif (B <= 1.12e+88)
		tmp = -(sqrt((2.0 * ((F * t_1) * (C + (A + hypot(B, (A - C))))))) / t_1);
	else
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9e-242], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1e-123], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.12e+88], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.9999999999999997e-242

   1. Initial program 16.0%

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

      1. associate-*l* 16.0%

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

      2. unpow2 16.0%

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

      3. +-commutative 16.0%

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

      4. unpow2 16.0%

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

      5. associate-*l* 16.0%

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

      6. unpow2 16.0%

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

   3. Simplified 16.0%

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

      1. *-un-lft-identity 16.0%

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

      2. associate-*l* 16.0%

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

      3. *-commutative 16.0%

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

      4. *-commutative 16.0%

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

      5. unpow2 16.0%

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

      6. hypot-udef 19.0%

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

      7. +-commutative 19.0%

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

      8. associate-+r+ 20.1%

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

   5. Applied egg-rr 20.1%

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

      1. *-lft-identity 20.1%

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

      2. associate-*r* 20.1%

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

      3. fma-neg 20.1%

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

      4. *-commutative 20.1%

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

      5. *-commutative 20.1%

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

      6. distribute-rgt-neg-in 20.1%

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

      7. *-commutative 20.1%

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

      8. metadata-eval 20.1%

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

      9. associate-*r* 20.1%

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

      10. associate-*r* 20.1%

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

      11. associate-*l* 17.4%

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

   7. Simplified 16.6%

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

   8. Taylor expanded in A around -inf 10.6%

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

      1. *-commutative 10.6%

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

   10. Simplified 10.6%

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

   if 8.9999999999999997e-242 < B < 1.0000000000000001e-123

   1. Initial program 8.6%

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

      1. associate-*l* 8.6%

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

      2. unpow2 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. associate-*l* 8.6%

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

      6. unpow2 8.6%

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

   3. Simplified 8.6%

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

      1. *-un-lft-identity 8.6%

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

      2. associate-*l* 8.6%

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

      3. *-commutative 8.6%

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

      4. *-commutative 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-udef 12.2%

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

      7. +-commutative 12.2%

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

      8. associate-+r+ 13.3%

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

   5. Applied egg-rr 13.3%

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

      1. *-lft-identity 13.3%

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

      2. associate-*r* 13.3%

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

      3. fma-neg 13.3%

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

      4. *-commutative 13.3%

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

      5. *-commutative 13.3%

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

      6. distribute-rgt-neg-in 13.3%

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

      7. *-commutative 13.3%

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

      8. metadata-eval 13.3%

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

      9. associate-*r* 13.3%

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

      10. associate-*r* 13.3%

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

      11. associate-*l* 13.3%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in A around inf 5.5%

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

      1. associate-+r+ 23.8%

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

      2. distribute-rgt1-in 23.8%

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

      3. metadata-eval 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 23.8%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 23.8%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.0000000000000001e-123 < B < 1.12000000000000006e88

   1. Initial program 36.1%

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

      1. associate-*l* 36.1%

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

      2. unpow2 36.1%

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

      3. +-commutative 36.1%

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

      4. unpow2 36.1%

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

      5. associate-*l* 36.1%

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

      6. unpow2 36.1%

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

   3. Simplified 36.1%

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

      1. distribute-frac-neg 36.1%

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

   5. Applied egg-rr 48.7%

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

   if 1.12000000000000006e88 < B

   1. Initial program 7.3%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in A around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. distribute-rgt-neg-in 17.0%

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

      3. *-commutative 17.0%

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

      4. unpow2 17.0%

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

      5. unpow2 17.0%

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

      6. hypot-def 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in C around 0 50.7%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{+88}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \end{array} \]

Alternative 7: 40.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 3.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 3.5e-303)
   (/
    (- (sqrt (* 2.0 (* (+ C (+ A (hypot B (- A C)))) (* F (* C (* A -4.0)))))))
    (- (* B B) (* 4.0 (* C A))))
   (if (<= F 1.85e+29)
     (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
     (- (* (sqrt 2.0) (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 3.5e-303) {
		tmp = -sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * (F * (C * (A * -4.0)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 1.85e+29) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 3.5e-303) {
		tmp = -Math.sqrt((2.0 * ((C + (A + Math.hypot(B, (A - C)))) * (F * (C * (A * -4.0)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 1.85e+29) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = -(Math.sqrt(2.0) * Math.sqrt((F / B)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 3.5e-303:
		tmp = -math.sqrt((2.0 * ((C + (A + math.hypot(B, (A - C)))) * (F * (C * (A * -4.0)))))) / ((B * B) - (4.0 * (C * A)))
	elif F <= 1.85e+29:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = -(math.sqrt(2.0) * math.sqrt((F / B)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 3.5e-303)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(C + Float64(A + hypot(B, Float64(A - C)))) * Float64(F * Float64(C * Float64(A * -4.0))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (F <= 1.85e+29)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 3.5e-303)
		tmp = -sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * (F * (C * (A * -4.0)))))) / ((B * B) - (4.0 * (C * A)));
	elseif (F <= 1.85e+29)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 3.5e-303], 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 * N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e+29], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < 3.5e-303

   1. Initial program 22.8%

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

      1. associate-*l* 22.8%

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

      2. unpow2 22.8%

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

      3. +-commutative 22.8%

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

      4. unpow2 22.8%

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

      5. associate-*l* 22.8%

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

      6. unpow2 22.8%

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

   3. Simplified 22.8%

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

      1. distribute-frac-neg 22.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in B around 0 32.8%

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

      1. associate-*r* 32.8%

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

      2. *-commutative 32.8%

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

      3. *-commutative 32.8%

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

   8. Simplified 32.8%

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

   if 3.5e-303 < F < 1.84999999999999987e29

   1. Initial program 18.6%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-rgt-neg-in 9.0%

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

      3. *-commutative 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. hypot-def 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in C around 0 19.9%

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

   if 1.84999999999999987e29 < F

   1. Initial program 12.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in B around inf 4.2%

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

   5. Taylor expanded in C around 0 18.2%

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

      1. associate-*r* 18.2%

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

      2. mul-1-neg 18.2%

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

   7. Simplified 18.2%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 8: 32.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq 6.8 \cdot 10^{-242}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot t_0\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0)))
        (t_1 (- (* B B) (* 4.0 (* C A))))
        (t_2 (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)))
   (if (<= B 6.8e-242)
     t_2
     (if (<= B 3.3e-78)
       (/ (- (sqrt (* 2.0 (* F (* (* 2.0 A) t_0))))) t_1)
       (if (<= B 8.6e-48)
         t_2
         (if (<= B 3.2e-24)
           (/ (- (sqrt (* (* F C) (* -16.0 (* A A))))) t_1)
           (- (* (sqrt 2.0) (sqrt (/ F B))))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double t_2 = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	double tmp;
	if (B <= 6.8e-242) {
		tmp = t_2;
	} else if (B <= 3.3e-78) {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 8.6e-48) {
		tmp = t_2;
	} else if (B <= 3.2e-24) {
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / t_1;
	} else {
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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) :: t_2
    real(8) :: tmp
    t_0 = (b * b) + ((c * a) * (-4.0d0))
    t_1 = (b * b) - (4.0d0 * (c * a))
    t_2 = -sqrt((2.0d0 * (f * (t_0 * (2.0d0 * c))))) / t_1
    if (b <= 6.8d-242) then
        tmp = t_2
    else if (b <= 3.3d-78) then
        tmp = -sqrt((2.0d0 * (f * ((2.0d0 * a) * t_0)))) / t_1
    else if (b <= 8.6d-48) then
        tmp = t_2
    else if (b <= 3.2d-24) then
        tmp = -sqrt(((f * c) * ((-16.0d0) * (a * a)))) / t_1
    else
        tmp = -(sqrt(2.0d0) * sqrt((f / b)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double t_2 = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	double tmp;
	if (B <= 6.8e-242) {
		tmp = t_2;
	} else if (B <= 3.3e-78) {
		tmp = -Math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	} else if (B <= 8.6e-48) {
		tmp = t_2;
	} else if (B <= 3.2e-24) {
		tmp = -Math.sqrt(((F * C) * (-16.0 * (A * A)))) / t_1;
	} else {
		tmp = -(Math.sqrt(2.0) * Math.sqrt((F / B)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	t_2 = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	tmp = 0
	if B <= 6.8e-242:
		tmp = t_2
	elif B <= 3.3e-78:
		tmp = -math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1
	elif B <= 8.6e-48:
		tmp = t_2
	elif B <= 3.2e-24:
		tmp = -math.sqrt(((F * C) * (-16.0 * (A * A)))) / t_1
	else:
		tmp = -(math.sqrt(2.0) * math.sqrt((F / B)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_2 = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1)
	tmp = 0.0
	if (B <= 6.8e-242)
		tmp = t_2;
	elseif (B <= 3.3e-78)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * t_0))))) / t_1);
	elseif (B <= 8.6e-48)
		tmp = t_2;
	elseif (B <= 3.2e-24)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * C) * Float64(-16.0 * Float64(A * A))))) / t_1);
	else
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	t_2 = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	tmp = 0.0;
	if (B <= 6.8e-242)
		tmp = t_2;
	elseif (B <= 3.3e-78)
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	elseif (B <= 8.6e-48)
		tmp = t_2;
	elseif (B <= 3.2e-24)
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / t_1;
	else
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, If[LessEqual[B, 6.8e-242], t$95$2, If[LessEqual[B, 3.3e-78], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 8.6e-48], t$95$2, If[LessEqual[B, 3.2e-24], N[((-N[Sqrt[N[(N[(F * C), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 6.8000000000000001e-242 or 3.29999999999999982e-78 < B < 8.6e-48

   1. Initial program 17.1%

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

      1. associate-*l* 17.1%

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

      2. unpow2 17.1%

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

      3. +-commutative 17.1%

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

      4. unpow2 17.1%

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

      5. associate-*l* 17.1%

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

      6. unpow2 17.1%

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

   3. Simplified 17.1%

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

      1. *-un-lft-identity 17.1%

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

      2. associate-*l* 17.1%

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

      3. *-commutative 17.1%

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

      4. *-commutative 17.1%

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

      5. unpow2 17.1%

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

      6. hypot-udef 21.1%

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

      7. +-commutative 21.1%

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

      8. associate-+r+ 22.3%

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

   5. Applied egg-rr 22.3%

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

      1. *-lft-identity 22.3%

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

      2. associate-*r* 22.3%

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

      3. fma-neg 22.3%

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

      4. *-commutative 22.3%

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

      5. *-commutative 22.3%

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

      6. distribute-rgt-neg-in 22.3%

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

      7. *-commutative 22.3%

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

      8. metadata-eval 22.3%

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

      9. associate-*r* 22.3%

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

      10. associate-*r* 22.3%

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

      11. associate-*l* 19.8%

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

   7. Simplified 18.9%

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

   8. Taylor expanded in A around -inf 13.9%

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

      1. *-commutative 13.9%

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

   10. Simplified 13.9%

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

   if 6.8000000000000001e-242 < B < 3.29999999999999982e-78

   1. Initial program 12.9%

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

      1. associate-*l* 12.9%

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

      2. unpow2 12.9%

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

      3. +-commutative 12.9%

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

      4. unpow2 12.9%

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

      5. associate-*l* 12.9%

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

      6. unpow2 12.9%

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

   3. Simplified 12.9%

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

      1. *-un-lft-identity 12.9%

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

      2. associate-*l* 12.9%

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

      3. *-commutative 12.9%

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

      4. *-commutative 12.9%

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

      5. unpow2 12.9%

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

      6. hypot-udef 18.3%

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

      7. +-commutative 18.3%

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

      8. associate-+r+ 19.2%

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

   5. Applied egg-rr 19.2%

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

      1. *-lft-identity 19.2%

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

      2. associate-*r* 19.2%

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

      3. fma-neg 19.2%

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

      4. *-commutative 19.2%

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

      5. *-commutative 19.2%

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

      6. distribute-rgt-neg-in 19.2%

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

      7. *-commutative 19.2%

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

      8. metadata-eval 19.2%

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

      9. associate-*r* 19.2%

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

      10. associate-*r* 19.2%

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

      11. associate-*l* 16.7%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in A around inf 7.5%

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

      1. associate-+r+ 22.5%

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

      2. distribute-rgt1-in 22.5%

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

      3. metadata-eval 22.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 22.5%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 22.5%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 8.6e-48 < B < 3.20000000000000012e-24

   1. Initial program 4.0%

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

      1. associate-*l* 4.0%

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

      2. unpow2 4.0%

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

      3. +-commutative 4.0%

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

      4. unpow2 4.0%

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

      5. associate-*l* 4.0%

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

      6. unpow2 4.0%

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

   3. Simplified 4.0%

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

   4. Taylor expanded in A around inf 2.2%

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

   5. Taylor expanded in A around inf 4.0%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 4.0%

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

      2. unpow2 4.0%

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

      3. *-commutative 4.0%

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

   7. Simplified 4.0%

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

   if 3.20000000000000012e-24 < B

   1. Initial program 18.7%

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

   3. Simplified 21.8%

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

   4. Taylor expanded in B around inf 14.9%

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

   5. Taylor expanded in C around 0 41.4%

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

      1. associate-*r* 41.4%

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

      2. mul-1-neg 41.4%

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

   7. Simplified 41.4%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 9: 37.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 3.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 3.2e-303)
   (/
    (- (sqrt (* 2.0 (* F (* (+ (* B B) (* (* C A) -4.0)) (* 2.0 C))))))
    (- (* B B) (* 4.0 (* C A))))
   (if (<= F 2.25e+29)
     (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
     (- (* (sqrt 2.0) (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 3.2e-303) {
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (2.0 * C))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2.25e+29) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (f <= 3.2d-303) then
        tmp = -sqrt((2.0d0 * (f * (((b * b) + ((c * a) * (-4.0d0))) * (2.0d0 * c))))) / ((b * b) - (4.0d0 * (c * a)))
    else if (f <= 2.25d+29) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * f))
    else
        tmp = -(sqrt(2.0d0) * sqrt((f / b)))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 3.2e-303) {
		tmp = -Math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (2.0 * C))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2.25e+29) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = -(Math.sqrt(2.0) * Math.sqrt((F / B)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 3.2e-303:
		tmp = -math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (2.0 * C))))) / ((B * B) - (4.0 * (C * A)))
	elif F <= 2.25e+29:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = -(math.sqrt(2.0) * math.sqrt((F / B)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 3.2e-303)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0)) * Float64(2.0 * C)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (F <= 2.25e+29)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(-Float64(sqrt(2.0) * sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 3.2e-303)
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (2.0 * C))))) / ((B * B) - (4.0 * (C * A)));
	elseif (F <= 2.25e+29)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = -(sqrt(2.0) * sqrt((F / B)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 3.2e-303], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.25e+29], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if F < 3.19999999999999991e-303

   1. Initial program 22.8%

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

      1. associate-*l* 22.8%

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

      2. unpow2 22.8%

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

      3. +-commutative 22.8%

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

      4. unpow2 22.8%

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

      5. associate-*l* 22.8%

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

      6. unpow2 22.8%

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

   3. Simplified 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. associate-*l* 22.8%

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

      3. *-commutative 22.8%

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

      4. *-commutative 22.8%

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

      5. unpow2 22.8%

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

      6. hypot-udef 32.8%

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

      7. +-commutative 32.8%

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

      8. associate-+r+ 32.8%

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

   5. Applied egg-rr 32.8%

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

      1. *-lft-identity 32.8%

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

      2. associate-*r* 32.8%

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

      3. fma-neg 32.8%

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

      4. *-commutative 32.8%

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

      5. *-commutative 32.8%

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

      6. distribute-rgt-neg-in 32.8%

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

      7. *-commutative 32.8%

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

      8. metadata-eval 32.8%

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

      9. associate-*r* 32.8%

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

      10. associate-*r* 32.8%

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

      11. associate-*l* 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in A around -inf 20.6%

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

      1. *-commutative 20.6%

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

   10. Simplified 20.6%

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

   if 3.19999999999999991e-303 < F < 2.2500000000000001e29

   1. Initial program 18.6%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-rgt-neg-in 9.0%

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

      3. *-commutative 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. hypot-def 22.0%

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

   6. Simplified 22.0%

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

   7. Taylor expanded in C around 0 19.9%

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

   if 2.2500000000000001e29 < F

   1. Initial program 12.5%

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

   3. Simplified 15.5%

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

   4. Taylor expanded in B around inf 4.2%

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

   5. Taylor expanded in C around 0 18.2%

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

      1. associate-*r* 18.2%

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

      2. mul-1-neg 18.2%

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

   7. Simplified 18.2%

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

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 10: 19.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot t_0\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= A -2.5e-121)
     (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)
     (if (<= A 1.15e-110)
       (/ (- (sqrt (* 2.0 (* F (* t_0 (+ A (+ B C))))))) t_1)
       (/ (- (sqrt (* 2.0 (* F (* (* 2.0 A) t_0))))) t_1)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -2.5e-121) {
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (A <= 1.15e-110) {
		tmp = -sqrt((2.0 * (F * (t_0 * (A + (B + C)))))) / t_1;
	} else {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = (b * b) + ((c * a) * (-4.0d0))
    t_1 = (b * b) - (4.0d0 * (c * a))
    if (a <= (-2.5d-121)) then
        tmp = -sqrt((2.0d0 * (f * (t_0 * (2.0d0 * c))))) / t_1
    else if (a <= 1.15d-110) then
        tmp = -sqrt((2.0d0 * (f * (t_0 * (a + (b + c)))))) / t_1
    else
        tmp = -sqrt((2.0d0 * (f * ((2.0d0 * a) * t_0)))) / t_1
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -2.5e-121) {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (A <= 1.15e-110) {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (A + (B + C)))))) / t_1;
	} else {
		tmp = -Math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if A <= -2.5e-121:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	elif A <= 1.15e-110:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (A + (B + C)))))) / t_1
	else:
		tmp = -math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (A <= -2.5e-121)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1);
	elseif (A <= 1.15e-110)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(A + Float64(B + C))))))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * t_0))))) / t_1);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (A <= -2.5e-121)
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	elseif (A <= 1.15e-110)
		tmp = -sqrt((2.0 * (F * (t_0 * (A + (B + C)))))) / t_1;
	else
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.5e-121], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[A, 1.15e-110], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -2.49999999999999995e-121

   1. Initial program 11.0%

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

      1. associate-*l* 11.0%

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

      2. unpow2 11.0%

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

      3. +-commutative 11.0%

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

      4. unpow2 11.0%

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

      5. associate-*l* 11.0%

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

      6. unpow2 11.0%

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

   3. Simplified 11.0%

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

      1. *-un-lft-identity 11.0%

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

      2. associate-*l* 11.0%

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

      3. *-commutative 11.0%

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

      4. *-commutative 11.0%

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

      5. unpow2 11.0%

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

      6. hypot-udef 12.3%

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

      7. +-commutative 12.3%

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

      8. associate-+r+ 15.0%

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

   5. Applied egg-rr 15.0%

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

      1. *-lft-identity 15.0%

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

      2. associate-*r* 15.0%

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

      3. fma-neg 15.0%

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

      4. *-commutative 15.0%

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

      5. *-commutative 15.0%

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

      6. distribute-rgt-neg-in 15.0%

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

      7. *-commutative 15.0%

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

      8. metadata-eval 15.0%

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

      9. associate-*r* 15.0%

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

      10. associate-*r* 15.0%

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

      11. associate-*l* 13.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in A around -inf 21.6%

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

      1. *-commutative 21.6%

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

   10. Simplified 21.6%

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

   if -2.49999999999999995e-121 < A < 1.1500000000000001e-110

   1. Initial program 19.3%

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

      1. associate-*l* 19.3%

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

      2. unpow2 19.3%

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

      3. +-commutative 19.3%

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

      4. unpow2 19.3%

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

      5. associate-*l* 19.3%

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

      6. unpow2 19.3%

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

   3. Simplified 19.3%

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

      1. *-un-lft-identity 19.3%

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

      2. associate-*l* 19.3%

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

      3. *-commutative 19.3%

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

      4. *-commutative 19.3%

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

      5. unpow2 19.3%

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

      6. hypot-udef 23.2%

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

      7. +-commutative 23.2%

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

      8. associate-+r+ 23.2%

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

   5. Applied egg-rr 23.2%

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

      1. *-lft-identity 23.2%

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

      2. associate-*r* 23.2%

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

      3. fma-neg 23.2%

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

      4. *-commutative 23.2%

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

      5. *-commutative 23.2%

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

      6. distribute-rgt-neg-in 23.2%

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

      7. *-commutative 23.2%

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

      8. metadata-eval 23.2%

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

      9. associate-*r* 23.2%

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

      10. associate-*r* 23.2%

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

      11. associate-*l* 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in B around inf 9.9%

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

   if 1.1500000000000001e-110 < A

   1. Initial program 19.7%

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

      1. associate-*l* 19.7%

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

      2. unpow2 19.7%

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

      3. +-commutative 19.7%

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

      4. unpow2 19.7%

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

      5. associate-*l* 19.7%

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

      6. unpow2 19.7%

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

   3. Simplified 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. associate-*l* 19.7%

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

      3. *-commutative 19.7%

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

      4. *-commutative 19.7%

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

      5. unpow2 19.7%

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

      6. hypot-udef 25.8%

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

      7. +-commutative 25.8%

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

      8. associate-+r+ 25.8%

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

   5. Applied egg-rr 25.8%

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

      1. *-lft-identity 25.8%

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

      2. associate-*r* 25.8%

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

      3. fma-neg 25.8%

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

      4. *-commutative 25.8%

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

      5. *-commutative 25.8%

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

      6. distribute-rgt-neg-in 25.8%

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

      7. *-commutative 25.8%

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

      8. metadata-eval 25.8%

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

      9. associate-*r* 25.8%

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

      10. associate-*r* 25.8%

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

      11. associate-*l* 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in A around inf 14.6%

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

      1. associate-+r+ 22.7%

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

      2. distribute-rgt1-in 22.7%

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

      3. metadata-eval 22.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 22.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 22.7%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;A \leq 1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(A + \left(B + C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 11: 17.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 2.12 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot t_0\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -6.6e-128)
     (/ (- (sqrt (* 2.0 (* -8.0 (* F (* C (* A A))))))) t_1)
     (if (<= C 2.12e-50)
       (/ (- (sqrt (* 2.0 (* F (* B t_0))))) t_1)
       (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -6.6e-128) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_1;
	} else if (C <= 2.12e-50) {
		tmp = -sqrt((2.0 * (F * (B * t_0)))) / t_1;
	} else {
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = (b * b) + ((c * a) * (-4.0d0))
    t_1 = (b * b) - (4.0d0 * (c * a))
    if (c <= (-6.6d-128)) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (c * (a * a)))))) / t_1
    else if (c <= 2.12d-50) then
        tmp = -sqrt((2.0d0 * (f * (b * t_0)))) / t_1
    else
        tmp = -sqrt((2.0d0 * (f * (t_0 * (2.0d0 * c))))) / t_1
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -6.6e-128) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_1;
	} else if (C <= 2.12e-50) {
		tmp = -Math.sqrt((2.0 * (F * (B * t_0)))) / t_1;
	} else {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -6.6e-128:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_1
	elif C <= 2.12e-50:
		tmp = -math.sqrt((2.0 * (F * (B * t_0)))) / t_1
	else:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -6.6e-128)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(C * Float64(A * A))))))) / t_1);
	elseif (C <= 2.12e-50)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(B * t_0))))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -6.6e-128)
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_1;
	elseif (C <= 2.12e-50)
		tmp = -sqrt((2.0 * (F * (B * t_0)))) / t_1;
	else
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -6.6e-128], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[C, 2.12e-50], N[((-N[Sqrt[N[(2.0 * N[(F * N[(B * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -6.6e-128

   1. Initial program 7.1%

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

      1. associate-*l* 7.1%

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

      2. unpow2 7.1%

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

      3. +-commutative 7.1%

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

      4. unpow2 7.1%

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

      5. associate-*l* 7.1%

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

      6. unpow2 7.1%

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

   3. Simplified 7.1%

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

      1. *-un-lft-identity 7.1%

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

      2. associate-*l* 7.1%

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

      3. *-commutative 7.1%

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

      4. *-commutative 7.1%

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

      5. unpow2 7.1%

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

      6. hypot-udef 8.7%

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

      7. +-commutative 8.7%

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

      8. associate-+r+ 9.7%

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

   5. Applied egg-rr 9.7%

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

      1. *-lft-identity 9.7%

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

      2. associate-*r* 9.7%

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

      3. fma-neg 9.7%

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

      4. *-commutative 9.7%

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

      5. *-commutative 9.7%

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

      6. distribute-rgt-neg-in 9.7%

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

      7. *-commutative 9.7%

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

      8. metadata-eval 9.7%

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

      9. associate-*r* 9.7%

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

      10. associate-*r* 9.7%

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

      11. associate-*l* 8.7%

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

   7. Simplified 7.8%

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

   8. Taylor expanded in B around 0 18.6%

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

      1. associate-*r* 20.8%

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

      2. unpow2 20.8%

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

   10. Simplified 20.8%

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

   if -6.6e-128 < C < 2.1199999999999999e-50

   1. Initial program 25.6%

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

      1. associate-*l* 25.6%

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

      2. unpow2 25.6%

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

      3. +-commutative 25.6%

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

      4. unpow2 25.6%

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

      5. associate-*l* 25.6%

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

      6. unpow2 25.6%

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

   3. Simplified 25.6%

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

      1. *-un-lft-identity 25.6%

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

      2. associate-*l* 25.6%

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

      3. *-commutative 25.6%

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

      4. *-commutative 25.6%

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

      5. unpow2 25.6%

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

      6. hypot-udef 31.0%

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

      7. +-commutative 31.0%

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

      8. associate-+r+ 31.8%

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

   5. Applied egg-rr 31.8%

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

      1. *-lft-identity 31.8%

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

      2. associate-*r* 31.8%

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

      3. fma-neg 31.8%

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

      4. *-commutative 31.8%

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

      5. *-commutative 31.8%

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

      6. distribute-rgt-neg-in 31.8%

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

      7. *-commutative 31.8%

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

      8. metadata-eval 31.8%

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

      9. associate-*r* 31.8%

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

      10. associate-*r* 31.8%

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

      11. associate-*l* 25.6%

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

   7. Simplified 25.1%

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

   8. Taylor expanded in B around inf 8.0%

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

   if 2.1199999999999999e-50 < C

   1. Initial program 17.8%

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

      1. associate-*l* 17.8%

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

      2. unpow2 17.8%

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

      3. +-commutative 17.8%

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

      4. unpow2 17.8%

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

      5. associate-*l* 17.8%

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

      6. unpow2 17.8%

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

   3. Simplified 17.8%

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

      1. *-un-lft-identity 17.8%

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

      2. associate-*l* 17.8%

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

      3. *-commutative 17.8%

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

      4. *-commutative 17.8%

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

      5. unpow2 17.8%

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

      6. hypot-udef 22.6%

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

      7. +-commutative 22.6%

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

      8. associate-+r+ 23.4%

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

   5. Applied egg-rr 23.4%

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

      1. *-lft-identity 23.4%

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

      2. associate-*r* 23.4%

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

      3. fma-neg 23.4%

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

      4. *-commutative 23.4%

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

      5. *-commutative 23.4%

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

      6. distribute-rgt-neg-in 23.4%

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

      7. *-commutative 23.4%

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

      8. metadata-eval 23.4%

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

      9. associate-*r* 23.4%

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

      10. associate-*r* 23.4%

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

      11. associate-*l* 22.0%

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

   7. Simplified 21.4%

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

   8. Taylor expanded in A around -inf 22.7%

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

      1. *-commutative 22.7%

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

   10. Simplified 22.7%

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

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 2.12 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 12: 19.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(t_0 \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot t_0\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot t_0\right)\right)}}{t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= A -1.6e-146)
     (/ (- (sqrt (* 2.0 (* F (* t_0 (* 2.0 C)))))) t_1)
     (if (<= A 1.4e-108)
       (/ (- (sqrt (* 2.0 (* F (* B t_0))))) t_1)
       (/ (- (sqrt (* 2.0 (* F (* (* 2.0 A) t_0))))) t_1)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -1.6e-146) {
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (A <= 1.4e-108) {
		tmp = -sqrt((2.0 * (F * (B * t_0)))) / t_1;
	} else {
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 = (b * b) + ((c * a) * (-4.0d0))
    t_1 = (b * b) - (4.0d0 * (c * a))
    if (a <= (-1.6d-146)) then
        tmp = -sqrt((2.0d0 * (f * (t_0 * (2.0d0 * c))))) / t_1
    else if (a <= 1.4d-108) then
        tmp = -sqrt((2.0d0 * (f * (b * t_0)))) / t_1
    else
        tmp = -sqrt((2.0d0 * (f * ((2.0d0 * a) * t_0)))) / t_1
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -1.6e-146) {
		tmp = -Math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	} else if (A <= 1.4e-108) {
		tmp = -Math.sqrt((2.0 * (F * (B * t_0)))) / t_1;
	} else {
		tmp = -Math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if A <= -1.6e-146:
		tmp = -math.sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1
	elif A <= 1.4e-108:
		tmp = -math.sqrt((2.0 * (F * (B * t_0)))) / t_1
	else:
		tmp = -math.sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (A <= -1.6e-146)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(t_0 * Float64(2.0 * C)))))) / t_1);
	elseif (A <= 1.4e-108)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(B * t_0))))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(2.0 * A) * t_0))))) / t_1);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (A <= -1.6e-146)
		tmp = -sqrt((2.0 * (F * (t_0 * (2.0 * C))))) / t_1;
	elseif (A <= 1.4e-108)
		tmp = -sqrt((2.0 * (F * (B * t_0)))) / t_1;
	else
		tmp = -sqrt((2.0 * (F * ((2.0 * A) * t_0)))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.6e-146], N[((-N[Sqrt[N[(2.0 * N[(F * N[(t$95$0 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[A, 1.4e-108], N[((-N[Sqrt[N[(2.0 * N[(F * N[(B * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(2.0 * A), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -1.6e-146

   1. Initial program 10.8%

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

      1. associate-*l* 10.8%

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

      2. unpow2 10.8%

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

      3. +-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. associate-*l* 10.8%

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

      6. unpow2 10.8%

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

   3. Simplified 10.8%

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

      1. *-un-lft-identity 10.8%

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

      2. associate-*l* 10.8%

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

      3. *-commutative 10.8%

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

      4. *-commutative 10.8%

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

      5. unpow2 10.8%

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

      6. hypot-udef 12.1%

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

      7. +-commutative 12.1%

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

      8. associate-+r+ 14.7%

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

   5. Applied egg-rr 14.7%

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

      1. *-lft-identity 14.7%

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

      2. associate-*r* 14.7%

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

      3. fma-neg 14.7%

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

      4. *-commutative 14.7%

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

      5. *-commutative 14.7%

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

      6. distribute-rgt-neg-in 14.7%

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

      7. *-commutative 14.7%

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

      8. metadata-eval 14.7%

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

      9. associate-*r* 14.7%

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

      10. associate-*r* 14.7%

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

      11. associate-*l* 13.1%

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

   7. Simplified 11.1%

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

   8. Taylor expanded in A around -inf 21.1%

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

      1. *-commutative 21.1%

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

   10. Simplified 21.1%

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

   if -1.6e-146 < A < 1.4e-108

   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. Step-by-step derivation

      1. *-un-lft-identity 19.8%

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

      2. associate-*l* 19.8%

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

      3. *-commutative 19.8%

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

      4. *-commutative 19.8%

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

      5. unpow2 19.8%

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

      6. hypot-udef 23.8%

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

      7. +-commutative 23.8%

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

      8. associate-+r+ 23.8%

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

   5. Applied egg-rr 23.8%

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

      1. *-lft-identity 23.8%

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

      2. associate-*r* 23.8%

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

      3. fma-neg 23.8%

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

      4. *-commutative 23.8%

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

      5. *-commutative 23.8%

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

      6. distribute-rgt-neg-in 23.8%

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

      7. *-commutative 23.8%

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

      8. metadata-eval 23.8%

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

      9. associate-*r* 23.8%

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

      10. associate-*r* 23.8%

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

      11. associate-*l* 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in B around inf 9.4%

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

   if 1.4e-108 < A

   1. Initial program 19.7%

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

      1. associate-*l* 19.7%

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

      2. unpow2 19.7%

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

      3. +-commutative 19.7%

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

      4. unpow2 19.7%

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

      5. associate-*l* 19.7%

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

      6. unpow2 19.7%

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

   3. Simplified 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. associate-*l* 19.7%

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

      3. *-commutative 19.7%

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

      4. *-commutative 19.7%

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

      5. unpow2 19.7%

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

      6. hypot-udef 25.8%

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

      7. +-commutative 25.8%

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

      8. associate-+r+ 25.8%

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

   5. Applied egg-rr 25.8%

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

      1. *-lft-identity 25.8%

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

      2. associate-*r* 25.8%

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

      3. fma-neg 25.8%

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

      4. *-commutative 25.8%

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

      5. *-commutative 25.8%

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

      6. distribute-rgt-neg-in 25.8%

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

      7. *-commutative 25.8%

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

      8. metadata-eval 25.8%

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

      9. associate-*r* 25.8%

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

      10. associate-*r* 25.8%

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

      11. associate-*l* 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in A around inf 14.6%

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

      1. associate-+r+ 22.7%

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

      2. distribute-rgt1-in 22.7%

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

      3. metadata-eval 22.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} \cdot C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. mul0-lft 22.7%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{0} + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 22.7%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(0 + 2 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.6 \cdot 10^{-146}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(2 \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(2 \cdot A\right) \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 13: 17.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -2.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + 2 \cdot A\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= A -2.2e-114)
     (/ (- (sqrt (* 2.0 (* -8.0 (* F (* A (* C C))))))) t_0)
     (if (<= A 2.6e-111)
       (/ (- (sqrt (* 2.0 (* F (* B (+ (* B B) (* (* C A) -4.0))))))) t_0)
       (if (<= A 1.7e+126)
         (/ (- (sqrt (* 2.0 (* -8.0 (* F (* C (* A A))))))) t_0)
         (/ (- (sqrt (* -8.0 (* A (* (* F C) (+ C (* 2.0 A))))))) t_0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -2.2e-114) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	} else if (A <= 2.6e-111) {
		tmp = -sqrt((2.0 * (F * (B * ((B * B) + ((C * A) * -4.0)))))) / t_0;
	} else if (A <= 1.7e+126) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else {
		tmp = -sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 * (c * a))
    if (a <= (-2.2d-114)) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (a * (c * c)))))) / t_0
    else if (a <= 2.6d-111) then
        tmp = -sqrt((2.0d0 * (f * (b * ((b * b) + ((c * a) * (-4.0d0))))))) / t_0
    else if (a <= 1.7d+126) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (c * (a * a)))))) / t_0
    else
        tmp = -sqrt(((-8.0d0) * (a * ((f * c) * (c + (2.0d0 * a)))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (A <= -2.2e-114) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	} else if (A <= 2.6e-111) {
		tmp = -Math.sqrt((2.0 * (F * (B * ((B * B) + ((C * A) * -4.0)))))) / t_0;
	} else if (A <= 1.7e+126) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else {
		tmp = -Math.sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if A <= -2.2e-114:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0
	elif A <= 2.6e-111:
		tmp = -math.sqrt((2.0 * (F * (B * ((B * B) + ((C * A) * -4.0)))))) / t_0
	elif A <= 1.7e+126:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0
	else:
		tmp = -math.sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (A <= -2.2e-114)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(A * Float64(C * C))))))) / t_0);
	elseif (A <= 2.6e-111)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(B * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))))) / t_0);
	elseif (A <= 1.7e+126)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(C * Float64(A * A))))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(Float64(F * C) * Float64(C + Float64(2.0 * A))))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (A <= -2.2e-114)
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	elseif (A <= 2.6e-111)
		tmp = -sqrt((2.0 * (F * (B * ((B * B) + ((C * A) * -4.0)))))) / t_0;
	elseif (A <= 1.7e+126)
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	else
		tmp = -sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.2e-114], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[A, 2.6e-111], N[((-N[Sqrt[N[(2.0 * N[(F * N[(B * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[A, 1.7e+126], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(N[(F * C), $MachinePrecision] * N[(C + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -2.20000000000000011e-114

   1. Initial program 8.9%

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

      1. associate-*l* 8.9%

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

      2. unpow2 8.9%

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

      3. +-commutative 8.9%

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

      4. unpow2 8.9%

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

      5. associate-*l* 8.9%

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

      6. unpow2 8.9%

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

   3. Simplified 8.9%

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

      1. *-un-lft-identity 8.9%

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

      2. associate-*l* 8.9%

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

      3. *-commutative 8.9%

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

      4. *-commutative 8.9%

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

      5. unpow2 8.9%

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

      6. hypot-udef 9.0%

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

      7. +-commutative 9.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 11.8%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 11.8%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 11.8%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 11.8%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 11.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 11.8%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 10.2%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 8.0%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around -inf 17.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 17.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 17.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 17.6%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -2.20000000000000011e-114 < A < 2.59999999999999982e-111

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 21.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 21.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 21.0%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 21.0%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 21.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 21.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 21.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 26.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 26.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 26.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 26.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 26.1%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 26.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 26.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 26.1%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 23.7%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 23.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around inf 9.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{B}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 2.59999999999999982e-111 < A < 1.69999999999999995e126

   1. Initial program 28.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 28.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 28.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 28.7%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 28.7%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 28.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 28.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 28.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 29.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 29.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 29.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 29.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 29.1%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 29.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 29.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 29.1%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 28.2%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 28.2%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around 0 25.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 26.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 26.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 26.9%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.69999999999999995e126 < A

   1. Initial program 4.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 4.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 4.6%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 20.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around 0 15.2%

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(C + 2 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 16.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 14: 12.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -7.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.02 \cdot 10^{-144}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{elif}\;C \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -7.8e-221)
     (/ (- (sqrt (* (* F C) (* -16.0 (* A A))))) t_0)
     (if (<= C 1.02e-144)
       (* -2.0 (/ (pow (* F A) 0.5) B))
       (if (<= C 6.2e-8)
         (* -2.0 (/ (sqrt (* F C)) B))
         (/ (- (sqrt (* 2.0 (* -8.0 (* F (* A (* C C))))))) t_0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -7.8e-221) {
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / t_0;
	} else if (C <= 1.02e-144) {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	} else if (C <= 6.2e-8) {
		tmp = -2.0 * (sqrt((F * C)) / B);
	} else {
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 * (c * a))
    if (c <= (-7.8d-221)) then
        tmp = -sqrt(((f * c) * ((-16.0d0) * (a * a)))) / t_0
    else if (c <= 1.02d-144) then
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    else if (c <= 6.2d-8) then
        tmp = (-2.0d0) * (sqrt((f * c)) / b)
    else
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (a * (c * c)))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -7.8e-221) {
		tmp = -Math.sqrt(((F * C) * (-16.0 * (A * A)))) / t_0;
	} else if (C <= 1.02e-144) {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	} else if (C <= 6.2e-8) {
		tmp = -2.0 * (Math.sqrt((F * C)) / B);
	} else {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -7.8e-221:
		tmp = -math.sqrt(((F * C) * (-16.0 * (A * A)))) / t_0
	elif C <= 1.02e-144:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	elif C <= 6.2e-8:
		tmp = -2.0 * (math.sqrt((F * C)) / B)
	else:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -7.8e-221)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * C) * Float64(-16.0 * Float64(A * A))))) / t_0);
	elseif (C <= 1.02e-144)
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	elseif (C <= 6.2e-8)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * C)) / B));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(A * Float64(C * C))))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -7.8e-221)
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / t_0;
	elseif (C <= 1.02e-144)
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	elseif (C <= 6.2e-8)
		tmp = -2.0 * (sqrt((F * C)) / B);
	else
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -7.8e-221], N[((-N[Sqrt[N[(N[(F * C), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1.02e-144], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 6.2e-8], N[(-2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -7.7999999999999997e-221

   1. Initial program 8.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 4.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around inf 16.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 16.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 16.3%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot \color{blue}{\left(A \cdot A\right)}\right) \cdot \left(C \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 16.3%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \color{blue}{\left(F \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 16.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -7.7999999999999997e-221 < C < 1.01999999999999997e-144

   1. Initial program 24.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 17.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 10.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 10.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

      2. mul-1-neg 10.7%

         \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

      3. unpow2 10.7%

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

      4. rem-square-sqrt 10.8%

         \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

      5. *-commutative 10.8%

         \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

   7. Simplified 10.8%

      \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

   8. Taylor expanded in B around 0 10.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 10.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      2. *-commutative 10.8%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

      3. *-rgt-identity 10.8%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

   10. Simplified 10.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]
   11. Step-by-step derivation

       1. pow1/2 11.1%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   12. Applied egg-rr 11.1%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   if 1.01999999999999997e-144 < C < 6.2e-8

   1. Initial program 31.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 31.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 31.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 31.1%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 31.1%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 31.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 31.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 31.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 33.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 33.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 34.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 34.2%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 34.2%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 34.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 34.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 34.2%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 29.0%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 28.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in C around inf 8.1%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(2 \cdot C + -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 8.1%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(2 \cdot C + \color{blue}{\left(-A\right)}\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 8.1%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \color{blue}{\left(2 \cdot C - A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 8.1%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(\color{blue}{C \cdot 2} - A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 8.1%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(C \cdot 2 - A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   11. Taylor expanded in B around inf 5.7%

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 5.7%

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity 5.7%

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

       3. *-commutative 5.7%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   13. Simplified 5.7%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if 6.2e-8 < C

   1. Initial program 16.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 16.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 16.5%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 16.5%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 16.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 16.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 16.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 22.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 22.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 23.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 23.2%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 23.2%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 23.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 23.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 23.2%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 21.5%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 20.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around -inf 19.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 18.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 18.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 18.0%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -7.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.02 \cdot 10^{-144}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{elif}\;C \leq 6.2 \cdot 10^{-8}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 15: 13.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -4.15 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.25 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + 2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -4.15e-35)
     (/ (- (sqrt (* 2.0 (* -8.0 (* F (* C (* A A))))))) t_0)
     (if (<= C 1.25e-66)
       (/ (- (sqrt (* -8.0 (* A (* (* F C) (+ C (* 2.0 A))))))) t_0)
       (/ (- (sqrt (* 2.0 (* -8.0 (* F (* A (* C C))))))) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -4.15e-35) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else if (C <= 1.25e-66) {
		tmp = -sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 * (c * a))
    if (c <= (-4.15d-35)) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (c * (a * a)))))) / t_0
    else if (c <= 1.25d-66) then
        tmp = -sqrt(((-8.0d0) * (a * ((f * c) * (c + (2.0d0 * a)))))) / t_0
    else
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (a * (c * c)))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -4.15e-35) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else if (C <= 1.25e-66) {
		tmp = -Math.sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	} else {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -4.15e-35:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0
	elif C <= 1.25e-66:
		tmp = -math.sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0
	else:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -4.15e-35)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(C * Float64(A * A))))))) / t_0);
	elseif (C <= 1.25e-66)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(Float64(F * C) * Float64(C + Float64(2.0 * A))))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(A * Float64(C * C))))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -4.15e-35)
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	elseif (C <= 1.25e-66)
		tmp = -sqrt((-8.0 * (A * ((F * C) * (C + (2.0 * A)))))) / t_0;
	else
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -4.15e-35], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1.25e-66], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(N[(F * C), $MachinePrecision] * N[(C + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -4.1499999999999998e-35

   1. Initial program 4.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 4.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 4.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 4.2%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 4.2%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 4.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 4.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 4.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 4.9%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 4.9%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 5.2%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 5.2%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 5.2%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 5.2%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 5.2%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 5.2%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 5.2%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 5.1%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around 0 19.5%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 22.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 22.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 22.4%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -4.1499999999999998e-35 < C < 1.2499999999999999e-66

   1. Initial program 24.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 24.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 15.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around 0 12.9%

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(C + 2 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.2499999999999999e-66 < C

   1. Initial program 18.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 18.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 18.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 18.4%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 18.4%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 18.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 18.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 18.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 23.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 23.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 23.8%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 23.8%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 23.8%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 23.8%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 23.8%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 23.8%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 22.4%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 21.9%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around -inf 16.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 15.2%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 15.2%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 15.2%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.15 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.25 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + 2 \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 16: 13.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -3.4 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -3.4e-297)
     (/ (- (sqrt (* 2.0 (* -8.0 (* F (* C (* A A))))))) t_0)
     (if (<= C 1.2e-142)
       (* -2.0 (/ (pow (* F A) 0.5) B))
       (/ (- (sqrt (* 2.0 (* -8.0 (* F (* A (* C C))))))) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -3.4e-297) {
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else if (C <= 1.2e-142) {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	} else {
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 * (c * a))
    if (c <= (-3.4d-297)) then
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (c * (a * a)))))) / t_0
    else if (c <= 1.2d-142) then
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    else
        tmp = -sqrt((2.0d0 * ((-8.0d0) * (f * (a * (c * c)))))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -3.4e-297) {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	} else if (C <= 1.2e-142) {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	} else {
		tmp = -Math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -3.4e-297:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0
	elif C <= 1.2e-142:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	else:
		tmp = -math.sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -3.4e-297)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(C * Float64(A * A))))))) / t_0);
	elseif (C <= 1.2e-142)
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(-8.0 * Float64(F * Float64(A * Float64(C * C))))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -3.4e-297)
		tmp = -sqrt((2.0 * (-8.0 * (F * (C * (A * A)))))) / t_0;
	elseif (C <= 1.2e-142)
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	else
		tmp = -sqrt((2.0 * (-8.0 * (F * (A * (C * C)))))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -3.4e-297], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1.2e-142], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(-8.0 * N[(F * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -3.39999999999999983e-297

   1. Initial program 9.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 9.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 9.6%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 9.6%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 9.6%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 9.6%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 9.6%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 9.6%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 12.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 12.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 13.5%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 13.5%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 13.5%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 13.5%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 13.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 13.5%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 12.5%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 11.8%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in B around 0 14.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 16.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 16.9%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 16.9%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -3.39999999999999983e-297 < C < 1.19999999999999994e-142

   1. Initial program 26.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 26.0%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 17.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 13.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 13.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

      2. mul-1-neg 13.7%

         \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

      3. unpow2 13.7%

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

      4. rem-square-sqrt 13.9%

         \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

      5. *-commutative 13.9%

         \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

   7. Simplified 13.9%

      \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

   8. Taylor expanded in B around 0 13.9%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 13.9%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      2. *-commutative 13.9%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

      3. *-rgt-identity 13.9%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

   10. Simplified 13.9%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]
   11. Step-by-step derivation

       1. pow1/2 14.2%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   12. Applied egg-rr 14.2%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   if 1.19999999999999994e-142 < C

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 22.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 22.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 22.1%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 22.1%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 22.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 22.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 22.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 26.8%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 26.8%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 27.6%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 27.6%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 27.6%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 27.6%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 27.6%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 27.6%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 24.6%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 24.0%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in A around -inf 15.0%

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 14.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 14.0%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 14.0%

       \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-8 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.4 \cdot 10^{-297}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 17: 12.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.25 \cdot 10^{-144}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.4e-217)
   (/ (- (sqrt (* (* F C) (* -16.0 (* A A))))) (- (* B B) (* 4.0 (* C A))))
   (if (<= C 1.25e-144)
     (* -2.0 (/ (pow (* F A) 0.5) B))
     (* -2.0 (/ (sqrt (* F C)) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.4e-217) {
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / ((B * B) - (4.0 * (C * A)));
	} else if (C <= 1.25e-144) {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	} else {
		tmp = -2.0 * (sqrt((F * C)) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (c <= (-1.4d-217)) then
        tmp = -sqrt(((f * c) * ((-16.0d0) * (a * a)))) / ((b * b) - (4.0d0 * (c * a)))
    else if (c <= 1.25d-144) then
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    else
        tmp = (-2.0d0) * (sqrt((f * c)) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.4e-217) {
		tmp = -Math.sqrt(((F * C) * (-16.0 * (A * A)))) / ((B * B) - (4.0 * (C * A)));
	} else if (C <= 1.25e-144) {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	} else {
		tmp = -2.0 * (Math.sqrt((F * C)) / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= -1.4e-217:
		tmp = -math.sqrt(((F * C) * (-16.0 * (A * A)))) / ((B * B) - (4.0 * (C * A)))
	elif C <= 1.25e-144:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	else:
		tmp = -2.0 * (math.sqrt((F * C)) / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1.4e-217)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * C) * Float64(-16.0 * Float64(A * A))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (C <= 1.25e-144)
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * C)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= -1.4e-217)
		tmp = -sqrt(((F * C) * (-16.0 * (A * A)))) / ((B * B) - (4.0 * (C * A)));
	elseif (C <= 1.25e-144)
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	else
		tmp = -2.0 * (sqrt((F * C)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, -1.4e-217], N[((-N[Sqrt[N[(N[(F * C), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.25e-144], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.4e-217

   1. Initial program 8.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 8.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 8.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 4.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around inf 16.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 16.3%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unpow2 16.3%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot \color{blue}{\left(A \cdot A\right)}\right) \cdot \left(C \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 16.3%

         \[\leadsto \frac{-\sqrt{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \color{blue}{\left(F \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 16.3%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot A\right)\right) \cdot \left(F \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.4e-217 < C < 1.2499999999999999e-144

   1. Initial program 24.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 24.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 17.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 10.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 10.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

      2. mul-1-neg 10.7%

         \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

      3. unpow2 10.7%

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

      4. rem-square-sqrt 10.8%

         \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

      5. *-commutative 10.8%

         \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

   7. Simplified 10.8%

      \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

   8. Taylor expanded in B around 0 10.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 10.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      2. *-commutative 10.8%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

      3. *-rgt-identity 10.8%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

   10. Simplified 10.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]
   11. Step-by-step derivation

       1. pow1/2 11.1%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   12. Applied egg-rr 11.1%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   if 1.2499999999999999e-144 < C

   1. Initial program 21.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 21.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 21.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 21.7%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 21.7%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 21.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 21.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 21.7%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 26.3%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 26.3%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 27.1%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 27.1%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 27.1%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 27.1%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 27.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 27.1%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 24.2%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 23.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in C around inf 15.4%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(2 \cdot C + -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 15.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(2 \cdot C + \color{blue}{\left(-A\right)}\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 15.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \color{blue}{\left(2 \cdot C - A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 15.4%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(\color{blue}{C \cdot 2} - A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 15.4%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(C \cdot 2 - A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   11. Taylor expanded in B around inf 6.2%

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 6.2%

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity 6.2%

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

       3. *-commutative 6.2%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   13. Simplified 6.2%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.25 \cdot 10^{-144}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 18: 7.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 3 \cdot 10^{-305}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= A 3e-305)
   (* -2.0 (/ (sqrt (* F C)) B))
   (* -2.0 (/ (pow (* F A) 0.5) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 3e-305) {
		tmp = -2.0 * (sqrt((F * C)) / B);
	} else {
		tmp = -2.0 * (pow((F * A), 0.5) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (a <= 3d-305) then
        tmp = (-2.0d0) * (sqrt((f * c)) / b)
    else
        tmp = (-2.0d0) * (((f * a) ** 0.5d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 3e-305) {
		tmp = -2.0 * (Math.sqrt((F * C)) / B);
	} else {
		tmp = -2.0 * (Math.pow((F * A), 0.5) / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if A <= 3e-305:
		tmp = -2.0 * (math.sqrt((F * C)) / B)
	else:
		tmp = -2.0 * (math.pow((F * A), 0.5) / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (A <= 3e-305)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * C)) / B));
	else
		tmp = Float64(-2.0 * Float64((Float64(F * A) ^ 0.5) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= 3e-305)
		tmp = -2.0 * (sqrt((F * C)) / B);
	else
		tmp = -2.0 * (((F * A) ^ 0.5) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[A, 3e-305], N[(-2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Power[N[(F * A), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 3.0000000000000001e-305

   1. Initial program 13.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 13.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 13.4%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 16.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 16.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 17.9%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 17.9%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 17.9%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 17.9%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 17.9%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 15.9%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 14.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in C around inf 8.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(2 \cdot C + -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(2 \cdot C + \color{blue}{\left(-A\right)}\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \color{blue}{\left(2 \cdot C - A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(\color{blue}{C \cdot 2} - A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 8.6%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(C \cdot 2 - A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   11. Taylor expanded in B around inf 5.3%

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 5.4%

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity 5.4%

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

       3. *-commutative 5.4%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   13. Simplified 5.4%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if 3.0000000000000001e-305 < A

   1. Initial program 19.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 19.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 12.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 5.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 5.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

      2. mul-1-neg 5.7%

         \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

      3. unpow2 5.7%

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

      4. rem-square-sqrt 5.8%

         \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

      5. *-commutative 5.8%

         \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

   7. Simplified 5.8%

      \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

   8. Taylor expanded in B around 0 5.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 5.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      2. *-commutative 5.8%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

      3. *-rgt-identity 5.8%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

   10. Simplified 5.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]
   11. Step-by-step derivation

       1. pow1/2 5.9%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]

   12. Applied egg-rr 5.9%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 3 \cdot 10^{-305}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{{\left(F \cdot A\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 19: 7.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;A \leq 7.5 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= A 7.5e-307)
   (* -2.0 (/ (sqrt (* F C)) B))
   (* -2.0 (/ (sqrt (* F A)) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 7.5e-307) {
		tmp = -2.0 * (sqrt((F * C)) / B);
	} else {
		tmp = -2.0 * (sqrt((F * A)) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive 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 (a <= 7.5d-307) then
        tmp = (-2.0d0) * (sqrt((f * c)) / b)
    else
        tmp = (-2.0d0) * (sqrt((f * a)) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= 7.5e-307) {
		tmp = -2.0 * (Math.sqrt((F * C)) / B);
	} else {
		tmp = -2.0 * (Math.sqrt((F * A)) / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if A <= 7.5e-307:
		tmp = -2.0 * (math.sqrt((F * C)) / B)
	else:
		tmp = -2.0 * (math.sqrt((F * A)) / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (A <= 7.5e-307)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * C)) / B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(F * A)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= 7.5e-307)
		tmp = -2.0 * (sqrt((F * C)) / B);
	else
		tmp = -2.0 * (sqrt((F * A)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[A, 7.5e-307], N[(-2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 7.5000000000000006e-307

   1. Initial program 13.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 13.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 13.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 13.4%

         \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*l* 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. unpow2 13.4%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. hypot-udef 16.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. +-commutative 16.0%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. associate-+r+ 17.9%

         \[\leadsto \frac{-1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 17.9%

      \[\leadsto \frac{-\color{blue}{1 \cdot \sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity 17.9%

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. associate-*r* 17.9%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. fma-neg 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      4. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      5. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      6. distribute-rgt-neg-in 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      7. *-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right)} \cdot \left(-4\right)\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      8. metadata-eval 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot \color{blue}{-4}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      9. associate-*r* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(A \cdot -4\right)}\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      10. associate-*r* 17.9%

          \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      11. associate-*l* 15.9%

          \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 14.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Taylor expanded in C around inf 8.6%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(2 \cdot C + -1 \cdot A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(2 \cdot C + \color{blue}{\left(-A\right)}\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      2. unsub-neg 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \color{blue}{\left(2 \cdot C - A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

      3. *-commutative 8.6%

         \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(\color{blue}{C \cdot 2} - A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   10. Simplified 8.6%

       \[\leadsto \frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(A + \left(C \cdot 2 - A\right)\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   11. Taylor expanded in B around inf 5.3%

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ 5.4%

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity 5.4%

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

       3. *-commutative 5.4%

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   13. Simplified 5.4%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if 7.5000000000000006e-307 < A

   1. Initial program 19.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 19.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 19.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around inf 12.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around 0 5.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 5.7%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

      2. mul-1-neg 5.7%

         \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

      3. unpow2 5.7%

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

      4. rem-square-sqrt 5.8%

         \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

      5. *-commutative 5.8%

         \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

   7. Simplified 5.8%

      \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

   8. Taylor expanded in B around 0 5.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 5.8%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

      2. *-commutative 5.8%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

      3. *-rgt-identity 5.8%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

   10. Simplified 5.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 7.5 \cdot 10^{-307}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot A}}{B}\\ \end{array} \]

Alternative 20: 5.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ -2 \cdot \frac{\sqrt{F \cdot A}}{B} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* F A)) B)))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((F * A)) / B);
}

Fortran

NOTE: B should be positive 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((f * a)) / b)
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((F * A)) / B);
}

Python

B = abs(B)
def code(A, B, C, F):
	return -2.0 * (math.sqrt((F * A)) / B)

Julia

B = abs(B)
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(F * A)) / B))
end

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((F * A)) / B);
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. associate-*l* 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. unpow2 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. +-commutative 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. unpow2 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. associate-*l* 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   6. unpow2 16.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

3. Simplified 16.8%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

4. Taylor expanded in A around inf 8.3%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

5. Taylor expanded in C around 0 3.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{A \cdot F}\right)} \]
6. Step-by-step derivation

   1. associate-*r* 3.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \cdot \sqrt{A \cdot F}} \]

   2. mul-1-neg 3.4%

      \[\leadsto \color{blue}{\left(-\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \cdot \sqrt{A \cdot F} \]

   3. unpow2 3.4%

      \[\leadsto \left(-\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \cdot \sqrt{A \cdot F} \]

   4. rem-square-sqrt 3.5%

      \[\leadsto \left(-\frac{\color{blue}{2}}{B}\right) \cdot \sqrt{A \cdot F} \]

   5. *-commutative 3.5%

      \[\leadsto \left(-\frac{2}{B}\right) \cdot \sqrt{\color{blue}{F \cdot A}} \]

7. Simplified 3.5%

   \[\leadsto \color{blue}{\left(-\frac{2}{B}\right) \cdot \sqrt{F \cdot A}} \]

8. Taylor expanded in B around 0 3.5%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 3.5%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

   2. *-commutative 3.5%

      \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot A}} \cdot 1}{B} \]

   3. *-rgt-identity 3.5%

      \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{F \cdot A}}}{B} \]

10. Simplified 3.5%

    \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot A}}{B}} \]

11. Final simplification 3.5%

    \[\leadsto -2 \cdot \frac{\sqrt{F \cdot A}}{B} \]

Reproduce

herbie shell --seed 2023240 
(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))))