ABCF->ab-angle a

Percentage Accurate: 18.4% → 51.3%

Time: 31.6s

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: 18.4% 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: 51.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{t_1} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (+ C (hypot C B))))
   (if (<= B -1.8e+50)
     (/ (* (sqrt (* F t_1)) (sqrt 2.0)) B)
     (if (<= B 4.7e-50)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (* (/ (sqrt 2.0) B) (* (sqrt t_1) (- (sqrt F))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = C + hypot(C, B);
	double tmp;
	if (B <= -1.8e+50) {
		tmp = (sqrt((F * t_1)) * sqrt(2.0)) / B;
	} else if (B <= 4.7e-50) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(t_1) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = C + Math.hypot(C, B);
	double tmp;
	if (B <= -1.8e+50) {
		tmp = (Math.sqrt((F * t_1)) * Math.sqrt(2.0)) / B;
	} else if (B <= 4.7e-50) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(t_1) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = C + math.hypot(C, B)
	tmp = 0
	if B <= -1.8e+50:
		tmp = (math.sqrt((F * t_1)) * math.sqrt(2.0)) / B
	elif B <= 4.7e-50:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(t_1) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(C + hypot(C, B))
	tmp = 0.0
	if (B <= -1.8e+50)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * sqrt(2.0)) / B);
	elseif (B <= 4.7e-50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(t_1) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = C + hypot(C, B);
	tmp = 0.0;
	if (B <= -1.8e+50)
		tmp = (sqrt((F * t_1)) * sqrt(2.0)) / B;
	elseif (B <= 4.7e-50)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(t_1) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.8e+50], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 4.7e-50], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.79999999999999993e50

   1. Initial program 12.4%

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

   3. Simplified 12.3%

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

      1. sqrt-prod 20.5%

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

      2. *-commutative 20.5%

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

      3. associate-+r+ 20.5%

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

      4. +-commutative 20.5%

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

      5. associate-+r+ 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in B around -inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. neg-mul-1 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in A around 0 19.1%

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

       1. associate-*r/ 19.2%

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

       2. *-commutative 19.2%

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

       3. +-commutative 19.2%

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

       4. unpow2 19.2%

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

       5. unpow2 19.2%

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

       6. hypot-def 49.0%

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

   11. Simplified 49.0%

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

   if -1.79999999999999993e50 < B < 4.7000000000000002e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 4.7000000000000002e-50 < B

   1. Initial program 17.6%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in A around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow2 25.1%

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

      4. unpow2 25.1%

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

   6. Simplified 25.1%

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

      1. sqrt-prod 29.2%

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

      2. hypot-udef 62.2%

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

   8. Applied egg-rr 62.2%

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

      1. hypot-def 29.2%

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

      2. unpow2 29.2%

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

      3. unpow2 29.2%

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

      4. +-commutative 29.2%

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

      5. unpow2 29.2%

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

      6. unpow2 29.2%

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

      7. hypot-def 62.2%

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

   10. Simplified 62.2%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2: 49.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 3.65 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -2.3e+50)
     (/ (* (sqrt (* F (+ C (hypot C B)))) (sqrt 2.0)) B)
     (if (<= B 3.65e-50)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (if (<= B 2.85e+113)
         (/ (* (sqrt 2.0) (- (sqrt (* F (+ C (hypot B C)))))) B)
         (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -2.3e+50) {
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	} else if (B <= 3.65e-50) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else if (B <= 2.85e+113) {
		tmp = (sqrt(2.0) * -sqrt((F * (C + hypot(B, C))))) / B;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -2.3e+50) {
		tmp = (Math.sqrt((F * (C + Math.hypot(C, B)))) * Math.sqrt(2.0)) / B;
	} else if (B <= 3.65e-50) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else if (B <= 2.85e+113) {
		tmp = (Math.sqrt(2.0) * -Math.sqrt((F * (C + Math.hypot(B, C))))) / B;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -2.3e+50:
		tmp = (math.sqrt((F * (C + math.hypot(C, B)))) * math.sqrt(2.0)) / B
	elif B <= 3.65e-50:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	elif B <= 2.85e+113:
		tmp = (math.sqrt(2.0) * -math.sqrt((F * (C + math.hypot(B, C))))) / B
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -2.3e+50)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * sqrt(2.0)) / B);
	elseif (B <= 3.65e-50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	elseif (B <= 2.85e+113)
		tmp = Float64(Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C)))))) / B);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -2.3e+50)
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	elseif (B <= 3.65e-50)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	elseif (B <= 2.85e+113)
		tmp = (sqrt(2.0) * -sqrt((F * (C + hypot(B, C))))) / B;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.3e+50], N[(N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 3.65e-50], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 2.85e+113], N[(N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / B), $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 < -2.29999999999999997e50

   1. Initial program 12.4%

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

   3. Simplified 12.3%

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

      1. sqrt-prod 20.5%

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

      2. *-commutative 20.5%

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

      3. associate-+r+ 20.5%

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

      4. +-commutative 20.5%

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

      5. associate-+r+ 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in B around -inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. neg-mul-1 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in A around 0 19.1%

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

       1. associate-*r/ 19.2%

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

       2. *-commutative 19.2%

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

       3. +-commutative 19.2%

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

       4. unpow2 19.2%

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

       5. unpow2 19.2%

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

       6. hypot-def 49.0%

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

   11. Simplified 49.0%

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

   if -2.29999999999999997e50 < B < 3.65000000000000018e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 3.65000000000000018e-50 < B < 2.8499999999999999e113

   1. Initial program 33.3%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in A around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. *-commutative 38.1%

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

      3. unpow2 38.1%

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

      4. unpow2 38.1%

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

   6. Simplified 38.1%

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

      1. associate-*l/ 38.1%

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

      2. hypot-udef 38.5%

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

   8. Applied egg-rr 38.5%

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

   if 2.8499999999999999e113 < B

   1. Initial program 3.2%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in A around 0 13.1%

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

      1. mul-1-neg 13.1%

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

      2. *-commutative 13.1%

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

      3. unpow2 13.1%

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

      4. unpow2 13.1%

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

   6. Simplified 13.1%

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

   7. Taylor expanded in C around 0 40.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.3 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 3.65 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 3: 49.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (/ (sqrt 2.0) B)))
   (if (<= B -1.15e+50)
     (/ (* (sqrt (* F (+ C (hypot C B)))) (sqrt 2.0)) B)
     (if (<= B 5e-50)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (if (<= B 8.2e+111)
         (* t_1 (- (sqrt (* F (+ C (hypot B C))))))
         (* t_1 (* (sqrt B) (- (sqrt F)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt(2.0) / B;
	double tmp;
	if (B <= -1.15e+50) {
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	} else if (B <= 5e-50) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else if (B <= 8.2e+111) {
		tmp = t_1 * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = t_1 * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = Math.sqrt(2.0) / B;
	double tmp;
	if (B <= -1.15e+50) {
		tmp = (Math.sqrt((F * (C + Math.hypot(C, B)))) * Math.sqrt(2.0)) / B;
	} else if (B <= 5e-50) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else if (B <= 8.2e+111) {
		tmp = t_1 * -Math.sqrt((F * (C + Math.hypot(B, C))));
	} else {
		tmp = t_1 * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = math.sqrt(2.0) / B
	tmp = 0
	if B <= -1.15e+50:
		tmp = (math.sqrt((F * (C + math.hypot(C, B)))) * math.sqrt(2.0)) / B
	elif B <= 5e-50:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	elif B <= 8.2e+111:
		tmp = t_1 * -math.sqrt((F * (C + math.hypot(B, C))))
	else:
		tmp = t_1 * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(sqrt(2.0) / B)
	tmp = 0.0
	if (B <= -1.15e+50)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * sqrt(2.0)) / B);
	elseif (B <= 5e-50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	elseif (B <= 8.2e+111)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(t_1 * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = sqrt(2.0) / B;
	tmp = 0.0;
	if (B <= -1.15e+50)
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	elseif (B <= 5e-50)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	elseif (B <= 8.2e+111)
		tmp = t_1 * -sqrt((F * (C + hypot(B, C))));
	else
		tmp = t_1 * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -1.15e+50], N[(N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 5e-50], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 8.2e+111], N[(t$95$1 * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.14999999999999998e50

   1. Initial program 12.4%

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

   3. Simplified 12.3%

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

      1. sqrt-prod 20.5%

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

      2. *-commutative 20.5%

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

      3. associate-+r+ 20.5%

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

      4. +-commutative 20.5%

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

      5. associate-+r+ 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in B around -inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. neg-mul-1 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in A around 0 19.1%

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

       1. associate-*r/ 19.2%

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

       2. *-commutative 19.2%

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

       3. +-commutative 19.2%

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

       4. unpow2 19.2%

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

       5. unpow2 19.2%

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

       6. hypot-def 49.0%

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

   11. Simplified 49.0%

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

   if -1.14999999999999998e50 < B < 4.99999999999999968e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 4.99999999999999968e-50 < B < 8.19999999999999973e111

   1. Initial program 33.3%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in A around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. *-commutative 38.1%

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

      3. distribute-rgt-neg-in 38.1%

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

      4. *-commutative 38.1%

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

      5. unpow2 38.1%

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

      6. unpow2 38.1%

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

      7. hypot-def 38.5%

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

   6. Simplified 38.5%

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

   if 8.19999999999999973e111 < B

   1. Initial program 3.2%

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

   3. Simplified 5.8%

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

   4. Taylor expanded in A around 0 13.1%

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

      1. mul-1-neg 13.1%

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

      2. *-commutative 13.1%

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

      3. unpow2 13.1%

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

      4. unpow2 13.1%

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

   6. Simplified 13.1%

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

   7. Taylor expanded in C around 0 40.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.15 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 4: 41.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ \mathbf{if}\;B \leq -2 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (sqrt (* 2.0 (* F t_0)))))
   (if (<= B -2e+152)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B -1.2e+50)
       (/ (* t_1 (- (sqrt (- C B)))) t_0)
       (if (<= B 5e-50)
         (/ (* t_1 (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A))))))) t_0)
         (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt((2.0 * (F * t_0)));
	double tmp;
	if (B <= -2e+152) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= -1.2e+50) {
		tmp = (t_1 * -sqrt((C - B))) / t_0;
	} else if (B <= 5e-50) {
		tmp = (t_1 * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    t_1 = sqrt((2.0d0 * (f * t_0)))
    if (b <= (-2d+152)) then
        tmp = 2.0d0 * (sqrt((f * c)) / b)
    else if (b <= (-1.2d+50)) then
        tmp = (t_1 * -sqrt((c - b))) / t_0
    else if (b <= 5d-50) then
        tmp = (t_1 * -sqrt((c + (c + ((-0.5d0) * ((b * b) / a)))))) / t_0
    else
        tmp = (sqrt(2.0d0) / b) * (sqrt(b) * -sqrt(f))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = Math.sqrt((2.0 * (F * t_0)));
	double tmp;
	if (B <= -2e+152) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else if (B <= -1.2e+50) {
		tmp = (t_1 * -Math.sqrt((C - B))) / t_0;
	} else if (B <= 5e-50) {
		tmp = (t_1 * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = math.sqrt((2.0 * (F * t_0)))
	tmp = 0
	if B <= -2e+152:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	elif B <= -1.2e+50:
		tmp = (t_1 * -math.sqrt((C - B))) / t_0
	elif B <= 5e-50:
		tmp = (t_1 * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = sqrt(Float64(2.0 * Float64(F * t_0)))
	tmp = 0.0
	if (B <= -2e+152)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= -1.2e+50)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C - B)))) / t_0);
	elseif (B <= 5e-50)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = sqrt((2.0 * (F * t_0)));
	tmp = 0.0;
	if (B <= -2e+152)
		tmp = 2.0 * (sqrt((F * C)) / B);
	elseif (B <= -1.2e+50)
		tmp = (t_1 * -sqrt((C - B))) / t_0;
	elseif (B <= 5e-50)
		tmp = (t_1 * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -2e+152], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.2e+50], N[(N[(t$95$1 * (-N[Sqrt[N[(C - B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 5e-50], N[(N[(t$95$1 * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -2.0000000000000001e152

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 6.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. 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%

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

   7. Simplified 6.2%

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

   if -2.0000000000000001e152 < B < -1.2000000000000001e50

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. unpow2 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l* 33.2%

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

      6. unpow2 33.2%

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

   3. Simplified 33.2%

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

      1. sqrt-prod 50.7%

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

      2. *-commutative 50.7%

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

      3. *-commutative 50.7%

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

      4. associate-+l+ 50.7%

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

      5. unpow2 50.7%

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

      6. hypot-udef 54.9%

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

      7. associate-+r+ 54.9%

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

      8. +-commutative 54.9%

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

      9. associate-+r+ 54.9%

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

   5. Applied egg-rr 54.9%

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

   6. Taylor expanded in B around -inf 43.3%

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

      1. mul-1-neg 43.3%

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

   8. Simplified 43.3%

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

   if -1.2000000000000001e50 < B < 4.99999999999999968e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 4.99999999999999968e-50 < B

   1. Initial program 17.6%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in A around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow2 25.1%

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

      4. unpow2 25.1%

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

   6. Simplified 25.1%

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

   7. Taylor expanded in C around 0 36.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 55.4%

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

   9. Applied egg-rr 55.4%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C - B}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5: 46.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B)) (t_1 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -3.1e+50)
     (* t_0 (sqrt (* F (+ A (hypot A B)))))
     (if (<= B 6.1e-50)
       (/
        (*
         (sqrt (* 2.0 (* F t_1)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_1)
       (* t_0 (* (sqrt B) (- (sqrt F))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(2.0) / B;
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.1e+50) {
		tmp = t_0 * sqrt((F * (A + hypot(A, B))));
	} else if (B <= 6.1e-50) {
		tmp = (sqrt((2.0 * (F * t_1))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	} else {
		tmp = t_0 * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B;
	double t_1 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.1e+50) {
		tmp = t_0 * Math.sqrt((F * (A + Math.hypot(A, B))));
	} else if (B <= 6.1e-50) {
		tmp = (Math.sqrt((2.0 * (F * t_1))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	} else {
		tmp = t_0 * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt(2.0) / B
	t_1 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -3.1e+50:
		tmp = t_0 * math.sqrt((F * (A + math.hypot(A, B))))
	elif B <= 6.1e-50:
		tmp = (math.sqrt((2.0 * (F * t_1))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1
	else:
		tmp = t_0 * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(2.0) / B)
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -3.1e+50)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(A + hypot(A, B)))));
	elseif (B <= 6.1e-50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_1))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_1);
	else
		tmp = Float64(t_0 * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt(2.0) / B;
	t_1 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -3.1e+50)
		tmp = t_0 * sqrt((F * (A + hypot(A, B))));
	elseif (B <= 6.1e-50)
		tmp = (sqrt((2.0 * (F * t_1))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_1;
	else
		tmp = t_0 * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.1e+50], N[(t$95$0 * N[Sqrt[N[(F * N[(A + N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.1e-50], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -3.10000000000000003e50

   1. Initial program 12.4%

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

   3. Simplified 12.3%

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

      1. sqrt-prod 20.5%

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

      2. *-commutative 20.5%

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

      3. associate-+r+ 20.5%

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

      4. +-commutative 20.5%

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

      5. associate-+r+ 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in B around -inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. neg-mul-1 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in C around 0 22.1%

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

       1. *-commutative 22.1%

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

       2. +-commutative 22.1%

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

       3. unpow2 22.1%

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

       4. unpow2 22.1%

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

       5. hypot-def 53.6%

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

   11. Simplified 53.6%

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

   if -3.10000000000000003e50 < B < 6.0999999999999996e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 6.0999999999999996e-50 < B

   1. Initial program 17.6%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in A around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow2 25.1%

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

      4. unpow2 25.1%

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

   6. Simplified 25.1%

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

   7. Taylor expanded in C around 0 36.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 55.4%

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

   9. Applied egg-rr 55.4%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 6: 48.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -1.2e+50)
     (/ (* (sqrt (* F (+ C (hypot C B)))) (sqrt 2.0)) B)
     (if (<= B 4.7e-50)
       (/
        (*
         (sqrt (* 2.0 (* F t_0)))
         (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))))
        t_0)
       (* (/ (sqrt 2.0) B) (* (sqrt B) (- (sqrt F))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -1.2e+50) {
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	} else if (B <= 4.7e-50) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -1.2e+50) {
		tmp = (Math.sqrt((F * (C + Math.hypot(C, B)))) * Math.sqrt(2.0)) / B;
	} else if (B <= 4.7e-50) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (Math.sqrt(B) * -Math.sqrt(F));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -1.2e+50:
		tmp = (math.sqrt((F * (C + math.hypot(C, B)))) * math.sqrt(2.0)) / B
	elif B <= 4.7e-50:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B) * (math.sqrt(B) * -math.sqrt(F))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -1.2e+50)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * sqrt(2.0)) / B);
	elseif (B <= 4.7e-50)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(B) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -1.2e+50)
		tmp = (sqrt((F * (C + hypot(C, B)))) * sqrt(2.0)) / B;
	elseif (B <= 4.7e-50)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = (sqrt(2.0) / B) * (sqrt(B) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.2e+50], N[(N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 4.7e-50], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[B], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.2000000000000001e50

   1. Initial program 12.4%

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

   3. Simplified 12.3%

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

      1. sqrt-prod 20.5%

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

      2. *-commutative 20.5%

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

      3. associate-+r+ 20.5%

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

      4. +-commutative 20.5%

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

      5. associate-+r+ 20.5%

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

   5. Applied egg-rr 20.5%

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

   6. Taylor expanded in B around -inf 27.0%

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

      1. associate-*r* 27.0%

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

      2. *-commutative 27.0%

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

      3. neg-mul-1 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in A around 0 19.1%

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

       1. associate-*r/ 19.2%

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

       2. *-commutative 19.2%

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

       3. +-commutative 19.2%

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

       4. unpow2 19.2%

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

       5. unpow2 19.2%

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

       6. hypot-def 49.0%

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

   11. Simplified 49.0%

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

   if -1.2000000000000001e50 < B < 4.7000000000000002e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 4.7000000000000002e-50 < B

   1. Initial program 17.6%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in A around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow2 25.1%

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

      4. unpow2 25.1%

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

   6. Simplified 25.1%

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

   7. Taylor expanded in C around 0 36.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 55.4%

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

   9. Applied egg-rr 55.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 7: 37.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (sqrt (* 2.0 (* F t_0)))))
   (if (<= B -3.7e+152)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B -1.2e+50)
       (/ (* t_1 (- (sqrt (- C B)))) t_0)
       (if (<= B 6.1e-50)
         (/ (* t_1 (- (sqrt (+ C (+ C (* -0.5 (/ (* B B) A))))))) t_0)
         (* (sqrt (/ F B)) (- (sqrt 2.0))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt((2.0 * (F * t_0)));
	double tmp;
	if (B <= -3.7e+152) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= -1.2e+50) {
		tmp = (t_1 * -sqrt((C - B))) / t_0;
	} else if (B <= 6.1e-50) {
		tmp = (t_1 * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    t_1 = sqrt((2.0d0 * (f * t_0)))
    if (b <= (-3.7d+152)) then
        tmp = 2.0d0 * (sqrt((f * c)) / b)
    else if (b <= (-1.2d+50)) then
        tmp = (t_1 * -sqrt((c - b))) / t_0
    else if (b <= 6.1d-50) then
        tmp = (t_1 * -sqrt((c + (c + ((-0.5d0) * ((b * b) / a)))))) / t_0
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = Math.sqrt((2.0 * (F * t_0)));
	double tmp;
	if (B <= -3.7e+152) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else if (B <= -1.2e+50) {
		tmp = (t_1 * -Math.sqrt((C - B))) / t_0;
	} else if (B <= 6.1e-50) {
		tmp = (t_1 * -Math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = math.sqrt((2.0 * (F * t_0)))
	tmp = 0
	if B <= -3.7e+152:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	elif B <= -1.2e+50:
		tmp = (t_1 * -math.sqrt((C - B))) / t_0
	elif B <= 6.1e-50:
		tmp = (t_1 * -math.sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = sqrt(Float64(2.0 * Float64(F * t_0)))
	tmp = 0.0
	if (B <= -3.7e+152)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= -1.2e+50)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C - B)))) / t_0);
	elseif (B <= 6.1e-50)
		tmp = Float64(Float64(t_1 * Float64(-sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = sqrt((2.0 * (F * t_0)));
	tmp = 0.0;
	if (B <= -3.7e+152)
		tmp = 2.0 * (sqrt((F * C)) / B);
	elseif (B <= -1.2e+50)
		tmp = (t_1 * -sqrt((C - B))) / t_0;
	elseif (B <= 6.1e-50)
		tmp = (t_1 * -sqrt((C + (C + (-0.5 * ((B * B) / A)))))) / t_0;
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -3.7e+152], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.2e+50], N[(N[(t$95$1 * (-N[Sqrt[N[(C - B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 6.1e-50], N[(N[(t$95$1 * (-N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -3.69999999999999996e152

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 6.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. 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%

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

   7. Simplified 6.2%

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

   if -3.69999999999999996e152 < B < -1.2000000000000001e50

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. unpow2 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l* 33.2%

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

      6. unpow2 33.2%

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

   3. Simplified 33.2%

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

      1. sqrt-prod 50.7%

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

      2. *-commutative 50.7%

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

      3. *-commutative 50.7%

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

      4. associate-+l+ 50.7%

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

      5. unpow2 50.7%

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

      6. hypot-udef 54.9%

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

      7. associate-+r+ 54.9%

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

      8. +-commutative 54.9%

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

      9. associate-+r+ 54.9%

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

   5. Applied egg-rr 54.9%

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

   6. Taylor expanded in B around -inf 43.3%

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

      1. mul-1-neg 43.3%

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

   8. Simplified 43.3%

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

   if -1.2000000000000001e50 < B < 6.0999999999999996e-50

   1. Initial program 20.4%

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

      1. associate-*l* 20.4%

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

      2. unpow2 20.4%

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

      3. +-commutative 20.4%

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

      4. unpow2 20.4%

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

      5. associate-*l* 20.4%

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

      6. unpow2 20.4%

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

   3. Simplified 20.4%

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

      1. sqrt-prod 23.2%

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

      2. *-commutative 23.2%

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

      3. *-commutative 23.2%

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

      4. associate-+l+ 24.1%

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

      5. unpow2 24.1%

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

      6. hypot-udef 41.2%

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

      7. associate-+r+ 39.3%

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

      8. +-commutative 39.3%

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

      9. associate-+r+ 40.9%

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

   5. Applied egg-rr 40.9%

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

   6. Taylor expanded in A around -inf 32.3%

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

      1. unpow2 32.3%

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

   8. Simplified 32.3%

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

   if 6.0999999999999996e-50 < B

   1. Initial program 17.6%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in A around 0 23.9%

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

      1. *-commutative 23.9%

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

      2. unpow2 23.9%

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

      3. unpow2 23.9%

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

      4. hypot-def 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in C around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. *-commutative 47.9%

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

      3. distribute-rgt-neg-in 47.9%

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

   9. Simplified 47.9%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C - B}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8: 38.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= B -3.7e+152)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B -2e+50)
       (/ (* (sqrt t_1) (- (sqrt (- C B)))) t_0)
       (if (<= B -4e-42)
         (/ (- (sqrt (* t_1 (fma 2.0 C (* -0.5 (/ (* B B) A)))))) t_0)
         (if (<= B 8.8e+21)
           (/
            (*
             (sqrt (* 2.0 (* F (+ (* B B) (* (* C A) -4.0)))))
             (- (sqrt (* C 2.0))))
            t_0)
           (* (sqrt (/ F B)) (- (sqrt 2.0)))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B <= -3.7e+152) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= -2e+50) {
		tmp = (sqrt(t_1) * -sqrt((C - B))) / t_0;
	} else if (B <= -4e-42) {
		tmp = -sqrt((t_1 * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else if (B <= 8.8e+21) {
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (B <= -3.7e+152)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= -2e+50)
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(Float64(C - B)))) / t_0);
	elseif (B <= -4e-42)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	elseif (B <= 8.8e+21)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.7e+152], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2e+50], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(C - B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, -4e-42], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 8.8e+21], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -3.69999999999999996e152

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 6.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. 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%

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

   7. Simplified 6.2%

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

   if -3.69999999999999996e152 < B < -2.0000000000000002e50

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. unpow2 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l* 33.2%

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

      6. unpow2 33.2%

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

   3. Simplified 33.2%

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

      1. sqrt-prod 50.7%

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

      2. *-commutative 50.7%

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

      3. *-commutative 50.7%

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

      4. associate-+l+ 50.7%

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

      5. unpow2 50.7%

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

      6. hypot-udef 54.9%

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

      7. associate-+r+ 54.9%

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

      8. +-commutative 54.9%

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

      9. associate-+r+ 54.9%

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

   5. Applied egg-rr 54.9%

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

   6. Taylor expanded in B around -inf 43.3%

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

      1. mul-1-neg 43.3%

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

   8. Simplified 43.3%

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

   if -2.0000000000000002e50 < B < -4.00000000000000015e-42

   1. Initial program 26.3%

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

      1. associate-*l* 26.3%

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

      2. unpow2 26.3%

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

      3. +-commutative 26.3%

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

      4. unpow2 26.3%

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

      5. associate-*l* 26.3%

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

      6. unpow2 26.3%

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

   3. Simplified 26.3%

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

   4. Taylor expanded in A around -inf 34.3%

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

      1. fma-def 34.3%

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

      2. unpow2 34.3%

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

   6. Simplified 34.3%

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

   if -4.00000000000000015e-42 < B < 8.8e21

   1. Initial program 20.6%

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

      1. associate-*l* 20.6%

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

      2. unpow2 20.6%

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

      3. +-commutative 20.6%

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

      4. unpow2 20.6%

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

      5. associate-*l* 20.6%

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

      6. unpow2 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in A around -inf 25.4%

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

      1. sqrt-prod 31.9%

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

      2. *-commutative 31.9%

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

      3. cancel-sign-sub-inv 31.9%

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

      4. metadata-eval 31.9%

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

      5. *-commutative 31.9%

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

   6. Applied egg-rr 31.9%

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

   if 8.8e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C - B}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9: 38.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -3.1e+144)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B -8.2e-30)
       (/ (- (sqrt (* (+ C (hypot B C)) (* 2.0 (* F t_0))))) t_0)
       (if (<= B 7.2e+21)
         (/
          (*
           (sqrt (* 2.0 (* F (+ (* B B) (* (* C A) -4.0)))))
           (- (sqrt (* C 2.0))))
          t_0)
         (* (sqrt (/ F B)) (- (sqrt 2.0))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.1e+144) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= -8.2e-30) {
		tmp = -sqrt(((C + hypot(B, C)) * (2.0 * (F * t_0)))) / t_0;
	} else if (B <= 7.2e+21) {
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.1e+144) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else if (B <= -8.2e-30) {
		tmp = -Math.sqrt(((C + Math.hypot(B, C)) * (2.0 * (F * t_0)))) / t_0;
	} else if (B <= 7.2e+21) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -Math.sqrt((C * 2.0))) / t_0;
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -3.1e+144:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	elif B <= -8.2e-30:
		tmp = -math.sqrt(((C + math.hypot(B, C)) * (2.0 * (F * t_0)))) / t_0
	elif B <= 7.2e+21:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -math.sqrt((C * 2.0))) / t_0
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -3.1e+144)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= -8.2e-30)
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + hypot(B, C)) * Float64(2.0 * Float64(F * t_0))))) / t_0);
	elseif (B <= 7.2e+21)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -3.1e+144)
		tmp = 2.0 * (sqrt((F * C)) / B);
	elseif (B <= -8.2e-30)
		tmp = -sqrt(((C + hypot(B, C)) * (2.0 * (F * t_0)))) / t_0;
	elseif (B <= 7.2e+21)
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / t_0;
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.1e+144], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -8.2e-30], N[((-N[Sqrt[N[(N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 7.2e+21], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -3.1000000000000002e144

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 6.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. 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%

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

   7. Simplified 6.2%

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

   if -3.1000000000000002e144 < B < -8.2000000000000007e-30

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. unpow2 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l* 33.2%

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

      6. unpow2 33.2%

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

   3. Simplified 33.2%

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

   4. Taylor expanded in A around 0 27.9%

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

      1. unpow2 27.9%

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

      2. unpow2 27.9%

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

      3. hypot-def 28.0%

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

   6. Simplified 28.0%

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

   if -8.2000000000000007e-30 < B < 7.2e21

   1. Initial program 20.0%

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

      1. associate-*l* 20.0%

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

      2. unpow2 20.0%

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

      3. +-commutative 20.0%

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

      4. unpow2 20.0%

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

      5. associate-*l* 20.0%

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

      6. unpow2 20.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in A around -inf 26.1%

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

      1. sqrt-prod 31.7%

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

      2. *-commutative 31.7%

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

      3. cancel-sign-sub-inv 31.7%

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

      4. metadata-eval 31.7%

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

      5. *-commutative 31.7%

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

   6. Applied egg-rr 31.7%

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

   if 7.2e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.1 \cdot 10^{+144}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10: 38.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -3.7e+152)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B -3e-28)
       (/ (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (- C B)))) t_0)
       (if (<= B 7.5e+21)
         (/
          (*
           (sqrt (* 2.0 (* F (+ (* B B) (* (* C A) -4.0)))))
           (- (sqrt (* C 2.0))))
          t_0)
         (* (sqrt (/ F B)) (- (sqrt 2.0))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.7e+152) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= -3e-28) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C - B))) / t_0;
	} else if (B <= 7.5e+21) {
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (b <= (-3.7d+152)) then
        tmp = 2.0d0 * (sqrt((f * c)) / b)
    else if (b <= (-3d-28)) then
        tmp = (sqrt((2.0d0 * (f * t_0))) * -sqrt((c - b))) / t_0
    else if (b <= 7.5d+21) then
        tmp = (sqrt((2.0d0 * (f * ((b * b) + ((c * a) * (-4.0d0)))))) * -sqrt((c * 2.0d0))) / t_0
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -3.7e+152) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else if (B <= -3e-28) {
		tmp = (Math.sqrt((2.0 * (F * t_0))) * -Math.sqrt((C - B))) / t_0;
	} else if (B <= 7.5e+21) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -Math.sqrt((C * 2.0))) / t_0;
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -3.7e+152:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	elif B <= -3e-28:
		tmp = (math.sqrt((2.0 * (F * t_0))) * -math.sqrt((C - B))) / t_0
	elif B <= 7.5e+21:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -math.sqrt((C * 2.0))) / t_0
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -3.7e+152)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= -3e-28)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C - B)))) / t_0);
	elseif (B <= 7.5e+21)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -3.7e+152)
		tmp = 2.0 * (sqrt((F * C)) / B);
	elseif (B <= -3e-28)
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C - B))) / t_0;
	elseif (B <= 7.5e+21)
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / t_0;
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.7e+152], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -3e-28], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C - B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 7.5e+21], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -3.69999999999999996e152

   1. Initial program 0.0%

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

      1. associate-*l* 0.0%

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

      2. unpow2 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*l* 0.0%

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

      6. unpow2 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around -inf 0.0%

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

   5. Taylor expanded in B around -inf 6.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. 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%

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

   7. Simplified 6.2%

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

   if -3.69999999999999996e152 < B < -3.00000000000000003e-28

   1. Initial program 33.2%

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

      1. associate-*l* 33.2%

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

      2. unpow2 33.2%

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

      3. +-commutative 33.2%

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

      4. unpow2 33.2%

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

      5. associate-*l* 33.2%

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

      6. unpow2 33.2%

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

   3. Simplified 33.2%

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

      1. sqrt-prod 43.9%

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

      2. *-commutative 43.9%

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

      3. *-commutative 43.9%

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

      4. associate-+l+ 44.3%

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

      5. unpow2 44.3%

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

      6. hypot-udef 50.0%

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

      7. associate-+r+ 49.2%

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

      8. +-commutative 49.2%

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

      9. associate-+r+ 49.3%

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

   5. Applied egg-rr 49.3%

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

   6. Taylor expanded in B around -inf 38.8%

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

      1. mul-1-neg 38.8%

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

   8. Simplified 38.8%

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

   if -3.00000000000000003e-28 < B < 7.5e21

   1. Initial program 20.0%

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

      1. associate-*l* 20.0%

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

      2. unpow2 20.0%

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

      3. +-commutative 20.0%

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

      4. unpow2 20.0%

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

      5. associate-*l* 20.0%

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

      6. unpow2 20.0%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in A around -inf 26.1%

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

      1. sqrt-prod 31.7%

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

      2. *-commutative 31.7%

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

      3. cancel-sign-sub-inv 31.7%

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

      4. metadata-eval 31.7%

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

      5. *-commutative 31.7%

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

   6. Applied egg-rr 31.7%

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

   if 7.5e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.7 \cdot 10^{+152}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C - B}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11: 36.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \frac{{\left(F \cdot C\right)}^{0.5}}{B}\\ \mathbf{elif}\;B \leq 7.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -2.2e+60)
   (* 2.0 (/ (pow (* F C) 0.5) B))
   (if (<= B 7.3e+21)
     (/
      (*
       (sqrt (* 2.0 (* F (+ (* B B) (* (* C A) -4.0)))))
       (- (sqrt (* C 2.0))))
      (- (* B B) (* 4.0 (* C A))))
     (* (sqrt (/ F B)) (- (sqrt 2.0))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2.2e+60) {
		tmp = 2.0 * (pow((F * C), 0.5) / B);
	} else if (B <= 7.3e+21) {
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / ((B * B) - (4.0 * (C * A)));
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-2.2d+60)) then
        tmp = 2.0d0 * (((f * c) ** 0.5d0) / b)
    else if (b <= 7.3d+21) then
        tmp = (sqrt((2.0d0 * (f * ((b * b) + ((c * a) * (-4.0d0)))))) * -sqrt((c * 2.0d0))) / ((b * b) - (4.0d0 * (c * a)))
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2.2e+60) {
		tmp = 2.0 * (Math.pow((F * C), 0.5) / B);
	} else if (B <= 7.3e+21) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -Math.sqrt((C * 2.0))) / ((B * B) - (4.0 * (C * A)));
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -2.2e+60:
		tmp = 2.0 * (math.pow((F * C), 0.5) / B)
	elif B <= 7.3e+21:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -math.sqrt((C * 2.0))) / ((B * B) - (4.0 * (C * A)))
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -2.2e+60)
		tmp = Float64(2.0 * Float64((Float64(F * C) ^ 0.5) / B));
	elseif (B <= 7.3e+21)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(C * 2.0)))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -2.2e+60)
		tmp = 2.0 * (((F * C) ^ 0.5) / B);
	elseif (B <= 7.3e+21)
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((C * 2.0))) / ((B * B) - (4.0 * (C * A)));
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -2.2e+60], N[(2.0 * N[(N[Power[N[(F * C), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 7.3e+21], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.19999999999999996e60

   1. Initial program 11.1%

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

      1. associate-*l* 11.1%

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

      2. unpow2 11.1%

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

      3. +-commutative 11.1%

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

      4. unpow2 11.1%

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

      5. associate-*l* 11.1%

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

      6. unpow2 11.1%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in A around -inf 0.4%

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

   5. Taylor expanded in B around -inf 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. *-commutative 4.8%

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

   7. Simplified 4.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
   8. Step-by-step derivation

      1. pow1/2 2.0%

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

   9. Applied egg-rr 5.0%

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

   if -2.19999999999999996e60 < B < 7.3e21

   1. Initial program 21.6%

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

      1. associate-*l* 21.6%

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

      2. unpow2 21.6%

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

      3. +-commutative 21.6%

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

      4. unpow2 21.6%

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

      5. associate-*l* 21.6%

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

      6. unpow2 21.6%

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

   3. Simplified 21.6%

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

   4. Taylor expanded in A around -inf 23.3%

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

      1. sqrt-prod 28.8%

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

      2. *-commutative 28.8%

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

      3. cancel-sign-sub-inv 28.8%

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

      4. metadata-eval 28.8%

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

      5. *-commutative 28.8%

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

   6. Applied egg-rr 28.8%

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

   if 7.3e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \frac{{\left(F \cdot C\right)}^{0.5}}{B}\\ \mathbf{elif}\;B \leq 7.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12: 36.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -9.8e+88)
     (* 2.0 (/ (sqrt (* F C)) B))
     (if (<= B 9.6e+21)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (* C 2.0)))) t_0)
       (* (sqrt (/ F B)) (- (sqrt 2.0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -9.8e+88) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else if (B <= 9.6e+21) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (b <= (-9.8d+88)) then
        tmp = 2.0d0 * (sqrt((f * c)) / b)
    else if (b <= 9.6d+21) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (c * 2.0d0))) / t_0
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -9.8e+88) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else if (B <= 9.6e+21) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -9.8e+88:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	elif B <= 9.6e+21:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -9.8e+88)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	elseif (B <= 9.6e+21)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -9.8e+88)
		tmp = 2.0 * (sqrt((F * C)) / B);
	elseif (B <= 9.6e+21)
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -9.8e+88], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 9.6e+21], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -9.8000000000000005e88

   1. Initial program 6.5%

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

      1. associate-*l* 6.5%

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

      2. unpow2 6.5%

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

      3. +-commutative 6.5%

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

      4. unpow2 6.5%

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

      5. associate-*l* 6.5%

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

      6. unpow2 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in A around -inf 0.3%

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

   5. Taylor expanded in B around -inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. *-commutative 5.2%

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

   7. Simplified 5.2%

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

   if -9.8000000000000005e88 < B < 9.6e21

   1. Initial program 22.7%

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

      1. associate-*l* 22.7%

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

      2. unpow2 22.7%

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

      3. +-commutative 22.7%

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

      4. unpow2 22.7%

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

      5. associate-*l* 22.7%

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

      6. unpow2 22.7%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in A around -inf 22.3%

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

   if 9.6e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13: 37.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= B -1.9e+52)
     (* 2.0 (/ (* (sqrt F) (sqrt C)) B))
     (if (<= B 7.8e+21)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (* C 2.0)))) t_0)
       (* (sqrt (/ F B)) (- (sqrt 2.0)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -1.9e+52) {
		tmp = 2.0 * ((sqrt(F) * sqrt(C)) / B);
	} else if (B <= 7.8e+21) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (b <= (-1.9d+52)) then
        tmp = 2.0d0 * ((sqrt(f) * sqrt(c)) / b)
    else if (b <= 7.8d+21) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (c * 2.0d0))) / t_0
    else
        tmp = sqrt((f / b)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (B <= -1.9e+52) {
		tmp = 2.0 * ((Math.sqrt(F) * Math.sqrt(C)) / B);
	} else if (B <= 7.8e+21) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = Math.sqrt((F / B)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if B <= -1.9e+52:
		tmp = 2.0 * ((math.sqrt(F) * math.sqrt(C)) / B)
	elif B <= 7.8e+21:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0
	else:
		tmp = math.sqrt((F / B)) * -math.sqrt(2.0)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (B <= -1.9e+52)
		tmp = Float64(2.0 * Float64(Float64(sqrt(F) * sqrt(C)) / B));
	elseif (B <= 7.8e+21)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (B <= -1.9e+52)
		tmp = 2.0 * ((sqrt(F) * sqrt(C)) / B);
	elseif (B <= 7.8e+21)
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	else
		tmp = sqrt((F / B)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.9e+52], N[(2.0 * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 7.8e+21], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.9e52

   1. Initial program 10.9%

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

      1. associate-*l* 10.9%

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

      2. unpow2 10.9%

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

      3. +-commutative 10.9%

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

      4. unpow2 10.9%

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

      5. associate-*l* 10.9%

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

      6. unpow2 10.9%

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

   3. Simplified 10.9%

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

   4. Taylor expanded in A around -inf 0.5%

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

   5. Taylor expanded in B around -inf 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. *-commutative 4.7%

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

   7. Simplified 4.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
   8. Step-by-step derivation

      1. sqrt-prod 6.3%

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

   9. Applied egg-rr 6.3%

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

   if -1.9e52 < B < 7.8e21

   1. Initial program 21.9%

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

      1. associate-*l* 21.9%

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

      2. unpow2 21.9%

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

      3. +-commutative 21.9%

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

      4. unpow2 21.9%

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

      5. associate-*l* 21.9%

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

      6. unpow2 21.9%

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

   3. Simplified 21.9%

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

   4. Taylor expanded in A around -inf 23.6%

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

   if 7.8e21 < B

   1. Initial program 14.7%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 21.8%

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

      1. *-commutative 21.8%

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

      2. unpow2 21.8%

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

      3. unpow2 21.8%

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

      4. hypot-def 24.3%

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

   6. Simplified 24.3%

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

   7. Taylor expanded in C around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. *-commutative 54.0%

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

      3. distribute-rgt-neg-in 54.0%

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

   9. Simplified 54.0%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F} \cdot \sqrt{C}}{B}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 14: 28.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{-{\left(2 \cdot \left(\left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{0.5}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* F C)) B)))
   (if (<= B -9.8e+88)
     (* 2.0 t_0)
     (if (<= B 1.3e+81)
       (/
        (- (pow (* 2.0 (* (* F (+ (* B B) (* (* C A) -4.0))) (* C 2.0))) 0.5))
        (- (* B B) (* 4.0 (* C A))))
       (* t_0 -2.0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * C)) / B;
	double tmp;
	if (B <= -9.8e+88) {
		tmp = 2.0 * t_0;
	} else if (B <= 1.3e+81) {
		tmp = -pow((2.0 * ((F * ((B * B) + ((C * A) * -4.0))) * (C * 2.0))), 0.5) / ((B * B) - (4.0 * (C * A)));
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((f * c)) / b
    if (b <= (-9.8d+88)) then
        tmp = 2.0d0 * t_0
    else if (b <= 1.3d+81) then
        tmp = -((2.0d0 * ((f * ((b * b) + ((c * a) * (-4.0d0)))) * (c * 2.0d0))) ** 0.5d0) / ((b * b) - (4.0d0 * (c * a)))
    else
        tmp = t_0 * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((F * C)) / B;
	double tmp;
	if (B <= -9.8e+88) {
		tmp = 2.0 * t_0;
	} else if (B <= 1.3e+81) {
		tmp = -Math.pow((2.0 * ((F * ((B * B) + ((C * A) * -4.0))) * (C * 2.0))), 0.5) / ((B * B) - (4.0 * (C * A)));
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((F * C)) / B
	tmp = 0
	if B <= -9.8e+88:
		tmp = 2.0 * t_0
	elif B <= 1.3e+81:
		tmp = -math.pow((2.0 * ((F * ((B * B) + ((C * A) * -4.0))) * (C * 2.0))), 0.5) / ((B * B) - (4.0 * (C * A)))
	else:
		tmp = t_0 * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(Float64(F * C)) / B)
	tmp = 0.0
	if (B <= -9.8e+88)
		tmp = Float64(2.0 * t_0);
	elseif (B <= 1.3e+81)
		tmp = Float64(Float64(-(Float64(2.0 * Float64(Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))) * Float64(C * 2.0))) ^ 0.5)) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	else
		tmp = Float64(t_0 * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((F * C)) / B;
	tmp = 0.0;
	if (B <= -9.8e+88)
		tmp = 2.0 * t_0;
	elseif (B <= 1.3e+81)
		tmp = -((2.0 * ((F * ((B * B) + ((C * A) * -4.0))) * (C * 2.0))) ^ 0.5) / ((B * B) - (4.0 * (C * A)));
	else
		tmp = t_0 * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -9.8e+88], N[(2.0 * t$95$0), $MachinePrecision], If[LessEqual[B, 1.3e+81], N[((-N[Power[N[(2.0 * N[(N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < -9.8000000000000005e88

   1. Initial program 6.5%

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

      1. associate-*l* 6.5%

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

      2. unpow2 6.5%

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

      3. +-commutative 6.5%

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

      4. unpow2 6.5%

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

      5. associate-*l* 6.5%

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

      6. unpow2 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in A around -inf 0.3%

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

   5. Taylor expanded in B around -inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. *-commutative 5.2%

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

   7. Simplified 5.2%

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

   if -9.8000000000000005e88 < B < 1.29999999999999996e81

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. unpow2 24.4%

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

      3. +-commutative 24.4%

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

      4. unpow2 24.4%

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

      5. associate-*l* 24.4%

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

      6. unpow2 24.4%

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

   3. Simplified 24.4%

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

   4. Taylor expanded in A around -inf 20.5%

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

      1. pow1/2 20.8%

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

      2. associate-*l* 20.8%

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

      3. *-commutative 20.8%

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

      4. cancel-sign-sub-inv 20.8%

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

      5. metadata-eval 20.8%

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

      6. *-commutative 20.8%

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

   6. Applied egg-rr 20.8%

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

   if 1.29999999999999996e81 < B

   1. Initial program 5.4%

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

      1. associate-*l* 5.4%

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

      2. unpow2 5.4%

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

      3. +-commutative 5.4%

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

      4. unpow2 5.4%

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

      5. associate-*l* 5.4%

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

      6. unpow2 5.4%

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

   3. Simplified 5.4%

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

   4. Taylor expanded in A around -inf 5.2%

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

   5. Taylor expanded in B around inf 7.4%

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

      1. associate-*r/ 7.4%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 7.4%

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

      3. *-commutative 7.4%

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

   7. Simplified 7.4%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{-{\left(2 \cdot \left(\left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right) \cdot \left(C \cdot 2\right)\right)\right)}^{0.5}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C}}{B} \cdot -2\\ \end{array} \]

Alternative 15: 28.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -4.6 \cdot 10^{+89}:\\ \;\;\;\;2 \cdot t_1\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* (* C A) -4.0))) (t_1 (/ (sqrt (* F C)) B)))
   (if (<= B -4.6e+89)
     (* 2.0 t_1)
     (if (<= B 2e+79)
       (/ (- (sqrt (* 2.0 (* (* F t_0) (* C 2.0))))) t_0)
       (* t_1 -2.0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = sqrt((F * C)) / B;
	double tmp;
	if (B <= -4.6e+89) {
		tmp = 2.0 * t_1;
	} else if (B <= 2e+79) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C * 2.0)))) / t_0;
	} else {
		tmp = t_1 * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) + ((c * a) * (-4.0d0))
    t_1 = sqrt((f * c)) / b
    if (b <= (-4.6d+89)) then
        tmp = 2.0d0 * t_1
    else if (b <= 2d+79) then
        tmp = -sqrt((2.0d0 * ((f * t_0) * (c * 2.0d0)))) / t_0
    else
        tmp = t_1 * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + ((C * A) * -4.0);
	double t_1 = Math.sqrt((F * C)) / B;
	double tmp;
	if (B <= -4.6e+89) {
		tmp = 2.0 * t_1;
	} else if (B <= 2e+79) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (C * 2.0)))) / t_0;
	} else {
		tmp = t_1 * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) + ((C * A) * -4.0)
	t_1 = math.sqrt((F * C)) / B
	tmp = 0
	if B <= -4.6e+89:
		tmp = 2.0 * t_1
	elif B <= 2e+79:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (C * 2.0)))) / t_0
	else:
		tmp = t_1 * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_1 = Float64(sqrt(Float64(F * C)) / B)
	tmp = 0.0
	if (B <= -4.6e+89)
		tmp = Float64(2.0 * t_1);
	elseif (B <= 2e+79)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C * 2.0))))) / t_0);
	else
		tmp = Float64(t_1 * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) + ((C * A) * -4.0);
	t_1 = sqrt((F * C)) / B;
	tmp = 0.0;
	if (B <= -4.6e+89)
		tmp = 2.0 * t_1;
	elseif (B <= 2e+79)
		tmp = -sqrt((2.0 * ((F * t_0) * (C * 2.0)))) / t_0;
	else
		tmp = t_1 * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -4.6e+89], N[(2.0 * t$95$1), $MachinePrecision], If[LessEqual[B, 2e+79], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(t$95$1 * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -4.5999999999999998e89

   1. Initial program 6.5%

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

      1. associate-*l* 6.5%

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

      2. unpow2 6.5%

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

      3. +-commutative 6.5%

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

      4. unpow2 6.5%

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

      5. associate-*l* 6.5%

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

      6. unpow2 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in A around -inf 0.3%

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

   5. Taylor expanded in B around -inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. *-commutative 5.2%

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

   7. Simplified 5.2%

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

   if -4.5999999999999998e89 < B < 1.99999999999999993e79

   1. Initial program 24.5%

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

      1. associate-*l* 24.5%

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

      2. unpow2 24.5%

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

      3. +-commutative 24.5%

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

      4. unpow2 24.5%

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

      5. associate-*l* 24.5%

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

      6. unpow2 24.5%

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

   3. Simplified 24.5%

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

   4. Taylor expanded in A around -inf 20.6%

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

      1. distribute-frac-neg 20.6%

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

      2. associate-*l* 20.6%

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

      3. *-commutative 20.6%

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

      4. cancel-sign-sub-inv 20.6%

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

      5. metadata-eval 20.6%

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

      6. *-commutative 20.6%

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

      7. cancel-sign-sub-inv 20.6%

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

      8. metadata-eval 20.6%

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

      9. *-commutative 20.6%

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

   6. Applied egg-rr 20.6%

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

   if 1.99999999999999993e79 < B

   1. Initial program 5.3%

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

      1. associate-*l* 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 5.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 5.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 5.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 7.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 7.3%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 7.3%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 7.3%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 7.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.6 \cdot 10^{+89}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right) \cdot \left(C \cdot 2\right)\right)}}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C}}{B} \cdot -2\\ \end{array} \]

Alternative 16: 28.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -1.24 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot t_1\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (/ (sqrt (* F C)) B)))
   (if (<= B -1.24e+88)
     (* 2.0 t_1)
     (if (<= B 2.65e+84)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (* C 2.0)))) t_0)
       (* t_1 -2.0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt((F * C)) / B;
	double tmp;
	if (B <= -1.24e+88) {
		tmp = 2.0 * t_1;
	} else if (B <= 2.65e+84) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = t_1 * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    t_1 = sqrt((f * c)) / b
    if (b <= (-1.24d+88)) then
        tmp = 2.0d0 * t_1
    else if (b <= 2.65d+84) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (c * 2.0d0))) / t_0
    else
        tmp = t_1 * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = Math.sqrt((F * C)) / B;
	double tmp;
	if (B <= -1.24e+88) {
		tmp = 2.0 * t_1;
	} else if (B <= 2.65e+84) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	} else {
		tmp = t_1 * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = math.sqrt((F * C)) / B
	tmp = 0
	if B <= -1.24e+88:
		tmp = 2.0 * t_1
	elif B <= 2.65e+84:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0
	else:
		tmp = t_1 * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(sqrt(Float64(F * C)) / B)
	tmp = 0.0
	if (B <= -1.24e+88)
		tmp = Float64(2.0 * t_1);
	elseif (B <= 2.65e+84)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(C * 2.0)))) / t_0);
	else
		tmp = Float64(t_1 * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = sqrt((F * C)) / B;
	tmp = 0.0;
	if (B <= -1.24e+88)
		tmp = 2.0 * t_1;
	elseif (B <= 2.65e+84)
		tmp = -sqrt(((2.0 * (F * t_0)) * (C * 2.0))) / t_0;
	else
		tmp = t_1 * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -1.24e+88], N[(2.0 * t$95$1), $MachinePrecision], If[LessEqual[B, 2.65e+84], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(t$95$1 * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -1.24e88

   1. Initial program 6.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 6.5%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 6.5%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 0.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 5.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 5.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 5.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 5.2%

         \[\leadsto 2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 5.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if -1.24e88 < B < 2.6500000000000001e84

   1. Initial program 24.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 24.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 24.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 20.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 2.6500000000000001e84 < B

   1. Initial program 5.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 5.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 5.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 5.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 7.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 7.4%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 7.4%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 7.4%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 7.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.24 \cdot 10^{+88}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+84}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C}}{B} \cdot -2\\ \end{array} \]

Alternative 17: 8.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq -1.2 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(F \cdot C\right)}^{0.5}}{B} \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B -1.2e-274)
   (* 2.0 (/ (sqrt (* F C)) B))
   (* (/ (pow (* F C) 0.5) B) -2.0)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.2e-274) {
		tmp = 2.0 * (sqrt((F * C)) / B);
	} else {
		tmp = (pow((F * C), 0.5) / B) * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= (-1.2d-274)) then
        tmp = 2.0d0 * (sqrt((f * c)) / b)
    else
        tmp = (((f * c) ** 0.5d0) / b) * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -1.2e-274) {
		tmp = 2.0 * (Math.sqrt((F * C)) / B);
	} else {
		tmp = (Math.pow((F * C), 0.5) / B) * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= -1.2e-274:
		tmp = 2.0 * (math.sqrt((F * C)) / B)
	else:
		tmp = (math.pow((F * C), 0.5) / B) * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= -1.2e-274)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) / B));
	else
		tmp = Float64(Float64((Float64(F * C) ^ 0.5) / B) * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= -1.2e-274)
		tmp = 2.0 * (sqrt((F * C)) / B);
	else
		tmp = (((F * C) ^ 0.5) / B) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, -1.2e-274], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(F * C), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -1.2e-274

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 13.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 4.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 4.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 4.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 4.2%

         \[\leadsto 2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 4.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if -1.2e-274 < B

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 15.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 5.1%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 5.1%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 5.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 5.1%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 5.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
   8. Step-by-step derivation

      1. pow1/2 5.5%

         \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]

   9. Applied egg-rr 5.5%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.2 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(F \cdot C\right)}^{0.5}}{B} \cdot -2\\ \end{array} \]

Alternative 18: 8.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{{\left(F \cdot C\right)}^{0.5}}{B}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (pow (* F C) 0.5) B)))
   (if (<= B -1.2e-274) (* 2.0 t_0) (* t_0 -2.0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = pow((F * C), 0.5) / B;
	double tmp;
	if (B <= -1.2e-274) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((f * c) ** 0.5d0) / b
    if (b <= (-1.2d-274)) then
        tmp = 2.0d0 * t_0
    else
        tmp = t_0 * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow((F * C), 0.5) / B;
	double tmp;
	if (B <= -1.2e-274) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.pow((F * C), 0.5) / B
	tmp = 0
	if B <= -1.2e-274:
		tmp = 2.0 * t_0
	else:
		tmp = t_0 * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64((Float64(F * C) ^ 0.5) / B)
	tmp = 0.0
	if (B <= -1.2e-274)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(t_0 * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = ((F * C) ^ 0.5) / B;
	tmp = 0.0;
	if (B <= -1.2e-274)
		tmp = 2.0 * t_0;
	else
		tmp = t_0 * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[N[(F * C), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -1.2e-274], N[(2.0 * t$95$0), $MachinePrecision], N[(t$95$0 * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -1.2e-274

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 13.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 4.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 4.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 4.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 4.2%

         \[\leadsto 2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 4.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
   8. Step-by-step derivation

      1. pow1/2 1.3%

         \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]

   9. Applied egg-rr 4.3%

      \[\leadsto 2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]

   if -1.2e-274 < B

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 15.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 5.1%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 5.1%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 5.1%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 5.1%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 5.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
   8. Step-by-step derivation

      1. pow1/2 5.5%

         \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]

   9. Applied egg-rr 5.5%

      \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot C\right)}^{0.5}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.2 \cdot 10^{-274}:\\ \;\;\;\;2 \cdot \frac{{\left(F \cdot C\right)}^{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(F \cdot C\right)}^{0.5}}{B} \cdot -2\\ \end{array} \]

Alternative 19: 8.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* F C)) B)))
   (if (<= B -2e-310) (* 2.0 t_0) (* t_0 -2.0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * C)) / B;
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((f * c)) / b
    if (b <= (-2d-310)) then
        tmp = 2.0d0 * t_0
    else
        tmp = t_0 * (-2.0d0)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((F * C)) / B;
	double tmp;
	if (B <= -2e-310) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * -2.0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((F * C)) / B
	tmp = 0
	if B <= -2e-310:
		tmp = 2.0 * t_0
	else:
		tmp = t_0 * -2.0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(sqrt(Float64(F * C)) / B)
	tmp = 0.0
	if (B <= -2e-310)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(t_0 * -2.0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((F * C)) / B;
	tmp = 0.0;
	if (B <= -2e-310)
		tmp = 2.0 * t_0;
	else
		tmp = t_0 * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -2e-310], N[(2.0 * t$95$0), $MachinePrecision], N[(t$95$0 * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -1.999999999999994e-310

   1. Initial program 17.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 17.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 17.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 13.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 4.0%

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 4.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 4.0%

         \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 4.0%

         \[\leadsto 2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 4.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

   if -1.999999999999994e-310 < B

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. associate-*l* 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*l* 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      6. unpow2 18.3%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Simplified 18.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   4. Taylor expanded in A around -inf 15.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around inf 5.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 5.3%

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

      2. *-rgt-identity 5.3%

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      3. *-commutative 5.3%

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 4.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot C}}{B} \cdot -2\\ \end{array} \]

Alternative 20: 5.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \frac{\sqrt{F \cdot C}}{B} \cdot -2 \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* (/ (sqrt (* F C)) B) -2.0))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return (sqrt((F * C)) / B) * -2.0;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt((f * c)) / b) * (-2.0d0)
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return (Math.sqrt((F * C)) / B) * -2.0;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return (math.sqrt((F * C)) / B) * -2.0

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(Float64(sqrt(Float64(F * C)) / B) * -2.0)
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = (sqrt((F * C)) / B) * -2.0;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

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. Taylor expanded in A around -inf 14.1%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

5. Taylor expanded in B around inf 3.1%

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 3.1%

      \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F} \cdot 1}{B}} \]

   2. *-rgt-identity 3.1%

      \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

   3. *-commutative 3.1%

      \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

7. Simplified 3.1%

   \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]

8. Final simplification 3.1%

   \[\leadsto \frac{\sqrt{F \cdot C}}{B} \cdot -2 \]

Reproduce

herbie shell --seed 2023182 
(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))))