ABCF->ab-angle a

Percentage Accurate: 19.3% → 45.3%

Time: 29.1s

Alternatives: 17

Speedup: 5.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.3% 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: 45.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.25 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_2}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{t_0}\right)\right) \cdot \left(-t_1\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{t_1}}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (sqrt (+ C (+ A (hypot B (- A C))))))
        (t_2 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A -2.25e+23)
     (/
      (- (sqrt (* (* 2.0 (* t_2 F)) (fma 2.0 C (* -0.5 (/ (* B B) A))))))
      t_2)
     (if (<= A 1.4e-304)
       (/ (* (* (sqrt 2.0) (* (sqrt F) (sqrt t_0))) (- t_1)) t_2)
       (/ (- (sqrt (* 2.0 (* F t_0)))) (/ (fma B B (* -4.0 (* A C))) t_1))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = sqrt((C + (A + hypot(B, (A - C)))));
	double t_2 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= -2.25e+23) {
		tmp = -sqrt(((2.0 * (t_2 * F)) * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_2;
	} else if (A <= 1.4e-304) {
		tmp = ((sqrt(2.0) * (sqrt(F) * sqrt(t_0))) * -t_1) / t_2;
	} else {
		tmp = -sqrt((2.0 * (F * t_0))) / (fma(B, B, (-4.0 * (A * C))) / t_1);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C)))))
	t_2 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -2.25e+23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_2);
	elseif (A <= 1.4e-304)
		tmp = Float64(Float64(Float64(sqrt(2.0) * Float64(sqrt(F) * sqrt(t_0))) * Float64(-t_1)) / t_2);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * t_0)))) / Float64(fma(B, B, Float64(-4.0 * Float64(A * C))) / t_1));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.25e+23], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[A, 1.4e-304], N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-t$95$1)), $MachinePrecision] / t$95$2), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -2.2499999999999999e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -2.2499999999999999e23 < A < 1.3999999999999999e-304

   1. Initial program 34.3%

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

      1. associate-*l* 34.3%

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

      2. unpow2 34.3%

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

      3. +-commutative 34.3%

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

      4. unpow2 34.3%

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

      5. associate-*l* 34.3%

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

      6. unpow2 34.3%

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

   3. Simplified 34.3%

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

      1. sqrt-prod 36.2%

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

      2. *-commutative 36.2%

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

      3. cancel-sign-sub-inv 36.2%

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

      4. metadata-eval 36.2%

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

      5. *-commutative 36.2%

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

      6. associate-+l+ 36.1%

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

      7. unpow2 36.1%

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

      8. hypot-udef 47.8%

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

      9. associate-+r+ 48.0%

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

      10. +-commutative 48.0%

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

   5. Applied egg-rr 48.3%

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

      1. sqrt-prod 48.3%

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

      2. fma-def 48.3%

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

      3. associate-*r* 48.3%

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

      4. *-commutative 48.3%

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

   7. Applied egg-rr 48.3%

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

      1. sqrt-prod 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. *-commutative 56.5%

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

   11. Simplified 56.5%

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

   if 1.3999999999999999e-304 < A

   1. Initial program 29.8%

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

      1. associate-*l* 29.8%

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

      2. unpow2 29.8%

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

      3. +-commutative 29.8%

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

      4. unpow2 29.8%

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

      5. associate-*l* 29.8%

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

      6. unpow2 29.8%

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

   3. Simplified 29.8%

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

      1. sqrt-prod 31.8%

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

      2. *-commutative 31.8%

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

      3. cancel-sign-sub-inv 31.8%

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

      4. metadata-eval 31.8%

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

      5. *-commutative 31.8%

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

      6. associate-+l+ 32.1%

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

      7. unpow2 32.1%

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

      8. hypot-udef 45.4%

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

      9. associate-+r+ 44.5%

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

      10. +-commutative 44.5%

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

   5. Applied egg-rr 44.5%

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

      1. sqrt-prod 44.5%

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

      2. fma-def 44.5%

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

      3. associate-*r* 44.5%

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

      4. *-commutative 44.5%

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

   7. Applied egg-rr 44.5%

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

      1. div-inv 44.4%

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

      2. distribute-lft-neg-in 44.4%

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

      3. sqrt-unprod 44.4%

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

      4. cancel-sign-sub-inv 44.4%

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

      5. metadata-eval 44.4%

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

   9. Applied egg-rr 44.4%

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

       1. associate-*r/ 44.5%

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

       2. *-rgt-identity 44.5%

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

       3. associate-/l* 44.5%

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

       4. *-commutative 44.5%

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

       5. fma-def 44.5%

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

       6. *-commutative 44.5%

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

       7. *-commutative 44.5%

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

   11. Simplified 44.5%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.25 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right) \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \end{array} \]

Alternative 2: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2 \cdot \left(F \cdot t_0\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0)))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A -1.55e+23)
     (/
      (- (sqrt (* (* 2.0 (* t_1 F)) (fma 2.0 C (* -0.5 (/ (* B B) A))))))
      t_1)
     (*
      (sqrt (+ C (+ A (hypot B (- A C)))))
      (/ (- (sqrt (* 2.0 (* F t_0)))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= -1.55e+23) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_1;
	} else {
		tmp = sqrt((C + (A + hypot(B, (A - C))))) * (-sqrt((2.0 * (F * t_0))) / t_0);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -1.55e+23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(Float64(-sqrt(Float64(2.0 * Float64(F * t_0)))) / t_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[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.55e+23], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.54999999999999985e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -1.54999999999999985e23 < A

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. unpow2 31.3%

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

      3. +-commutative 31.3%

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

      4. unpow2 31.3%

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

      5. associate-*l* 31.3%

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

      6. unpow2 31.3%

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

   3. Simplified 31.3%

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

      1. sqrt-prod 33.2%

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

      2. *-commutative 33.2%

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

      3. cancel-sign-sub-inv 33.2%

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

      4. metadata-eval 33.2%

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

      5. *-commutative 33.2%

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

      6. associate-+l+ 33.4%

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

      7. unpow2 33.4%

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

      8. hypot-udef 46.2%

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

      9. associate-+r+ 45.6%

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

      10. +-commutative 45.6%

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

   5. Applied egg-rr 45.7%

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

      1. sqrt-prod 45.7%

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

      2. fma-def 45.7%

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

      3. associate-*r* 45.7%

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

      4. *-commutative 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. div-inv 45.7%

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

      2. distribute-lft-neg-in 45.7%

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

      3. sqrt-unprod 45.6%

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

      4. cancel-sign-sub-inv 45.6%

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

      5. metadata-eval 45.6%

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

   9. Applied egg-rr 45.6%

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

       1. associate-*r/ 45.7%

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

       2. *-rgt-identity 45.7%

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

       3. associate-/l* 45.8%

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

       4. *-commutative 45.8%

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

       5. fma-def 45.8%

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

       6. *-commutative 45.8%

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

       7. *-commutative 45.8%

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

   11. Simplified 45.8%

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

       1. associate-/r/ 45.8%

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

       2. *-commutative 45.8%

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

       3. associate-*r* 45.8%

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

   13. Applied egg-rr 45.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 3: 42.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.38 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A -1.38e+23)
     (/
      (- (sqrt (* (* 2.0 (* t_0 F)) (fma 2.0 C (* -0.5 (/ (* B B) A))))))
      t_0)
     (/
      (- (sqrt (* 2.0 (* F (fma B B (* A (* C -4.0)))))))
      (/ (fma B B (* -4.0 (* A C))) (sqrt (+ C (+ A (hypot B (- A C))))))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= -1.38e+23) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else {
		tmp = -sqrt((2.0 * (F * fma(B, B, (A * (C * -4.0)))))) / (fma(B, B, (-4.0 * (A * C))) / sqrt((C + (A + hypot(B, (A - C))))));
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -1.38e+23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(A * Float64(C * -4.0))))))) / Float64(fma(B, B, Float64(-4.0 * Float64(A * C))) / sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C)))))));
	end
	return tmp
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.38e+23], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.38e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -1.38e23 < A

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. unpow2 31.3%

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

      3. +-commutative 31.3%

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

      4. unpow2 31.3%

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

      5. associate-*l* 31.3%

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

      6. unpow2 31.3%

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

   3. Simplified 31.3%

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

      1. sqrt-prod 33.2%

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

      2. *-commutative 33.2%

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

      3. cancel-sign-sub-inv 33.2%

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

      4. metadata-eval 33.2%

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

      5. *-commutative 33.2%

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

      6. associate-+l+ 33.4%

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

      7. unpow2 33.4%

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

      8. hypot-udef 46.2%

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

      9. associate-+r+ 45.6%

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

      10. +-commutative 45.6%

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

   5. Applied egg-rr 45.7%

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

      1. sqrt-prod 45.7%

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

      2. fma-def 45.7%

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

      3. associate-*r* 45.7%

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

      4. *-commutative 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. div-inv 45.7%

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

      2. distribute-lft-neg-in 45.7%

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

      3. sqrt-unprod 45.6%

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

      4. cancel-sign-sub-inv 45.6%

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

      5. metadata-eval 45.6%

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

   9. Applied egg-rr 45.6%

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

       1. associate-*r/ 45.7%

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

       2. *-rgt-identity 45.7%

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

       3. associate-/l* 45.8%

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

       4. *-commutative 45.8%

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

       5. fma-def 45.8%

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

       6. *-commutative 45.8%

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

       7. *-commutative 45.8%

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

   11. Simplified 45.8%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.38 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}\\ \end{array} \]

Alternative 4: 42.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A -1.6e+23)
     (/
      (- (sqrt (* (* 2.0 (* t_0 F)) (fma 2.0 C (* -0.5 (/ (* B B) A))))))
      t_0)
     (/
      (*
       (sqrt (+ C (+ A (hypot B (- A C)))))
       (- (sqrt (* 2.0 (* F (+ (* B B) (* -4.0 (* A C))))))))
      t_0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= -1.6e+23) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * ((B * B) + (-4.0 * (A * C))))))) / t_0;
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -1.6e+23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))))))) / t_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[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.6e+23], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.6e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -1.6e23 < A

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. unpow2 31.3%

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

      3. +-commutative 31.3%

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

      4. unpow2 31.3%

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

      5. associate-*l* 31.3%

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

      6. unpow2 31.3%

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

   3. Simplified 31.3%

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

      1. sqrt-prod 33.2%

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

      2. *-commutative 33.2%

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

      3. cancel-sign-sub-inv 33.2%

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

      4. metadata-eval 33.2%

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

      5. *-commutative 33.2%

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

      6. associate-+l+ 33.4%

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

      7. unpow2 33.4%

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

      8. hypot-udef 46.2%

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

      9. associate-+r+ 45.6%

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

      10. +-commutative 45.6%

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

   5. Applied egg-rr 45.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 5: 36.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(t_0 \cdot F\right)\\ \mathbf{if}\;A \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* t_0 F))))
   (if (<= A -1.55e+23)
     (/ (- (sqrt (* t_1 (fma 2.0 C (* -0.5 (/ (* B B) A)))))) t_0)
     (/ (- (sqrt (* t_1 (+ C (hypot C B))))) t_0))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (t_0 * F);
	double tmp;
	if (A <= -1.55e+23) {
		tmp = -sqrt((t_1 * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else {
		tmp = -sqrt((t_1 * (C + hypot(C, B)))) / t_0;
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	tmp = 0.0
	if (A <= -1.55e+23)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(C + hypot(C, B))))) / t_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[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.55e+23], 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], N[((-N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.54999999999999985e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -1.54999999999999985e23 < A

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. unpow2 31.3%

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

      3. +-commutative 31.3%

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

      4. unpow2 31.3%

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

      5. associate-*l* 31.3%

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

      6. unpow2 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in A around 0 25.2%

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

      1. +-commutative 25.2%

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

      2. unpow2 25.2%

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

      3. unpow2 25.2%

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

      4. hypot-def 30.0%

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

   6. Simplified 30.0%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 6: 36.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.25 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot t_0\right)\right)} \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (* B B) (* -4.0 (* A C)))) (t_1 (- (* B B) (* 4.0 (* A C)))))
   (if (<= A -2.25e+23)
     (/
      (- (sqrt (* (* 2.0 (* t_1 F)) (fma 2.0 C (* -0.5 (/ (* B B) A))))))
      t_1)
     (*
      (sqrt (* 2.0 (* (+ C (+ A (hypot B (- A C)))) (* F t_0))))
      (/ -1.0 t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) + (-4.0 * (A * C));
	double t_1 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (A <= -2.25e+23) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_1;
	} else {
		tmp = sqrt((2.0 * ((C + (A + hypot(B, (A - C)))) * (F * t_0)))) * (-1.0 / t_0);
	}
	return tmp;
}

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -2.25e+23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(C + Float64(A + hypot(B, Float64(A - C)))) * Float64(F * t_0)))) * Float64(-1.0 / t_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[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.25e+23], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if A < -2.2499999999999999e23

   1. Initial program 1.6%

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

      1. associate-*l* 1.6%

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

      2. unpow2 1.6%

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

      3. +-commutative 1.6%

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

      4. unpow2 1.6%

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

      5. associate-*l* 1.6%

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

      6. unpow2 1.6%

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

   3. Simplified 1.6%

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

   4. Taylor expanded in A around -inf 41.7%

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

      1. fma-def 41.7%

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

      2. unpow2 41.7%

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

   6. Simplified 41.7%

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

   if -2.2499999999999999e23 < A

   1. Initial program 31.3%

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

      1. associate-*l* 31.3%

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

      2. unpow2 31.3%

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

      3. +-commutative 31.3%

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

      4. unpow2 31.3%

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

      5. associate-*l* 31.3%

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

      6. unpow2 31.3%

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

   3. Simplified 31.3%

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

      1. div-inv 31.2%

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

   5. Applied egg-rr 37.5%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.25 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 7: 36.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(t_0 \cdot F\right)\\ \mathbf{if}\;C \leq -5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* t_0 F))))
   (if (<= C -5.8e-180)
     (/ (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0)))))))) t_0)
     (if (<= C 1.45e+87)
       (/ (- (sqrt (* t_1 (+ C (hypot C B))))) t_0)
       (/ (* (sqrt t_1) (- (sqrt (* 2.0 C)))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (t_0 * F);
	double tmp;
	if (C <= -5.8e-180) {
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (C <= 1.45e+87) {
		tmp = -sqrt((t_1 * (C + hypot(C, B)))) / t_0;
	} else {
		tmp = (sqrt(t_1) * -sqrt((2.0 * C))) / t_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 * (A * C));
	double t_1 = 2.0 * (t_0 * F);
	double tmp;
	if (C <= -5.8e-180) {
		tmp = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (C <= 1.45e+87) {
		tmp = -Math.sqrt((t_1 * (C + Math.hypot(C, B)))) / t_0;
	} else {
		tmp = (Math.sqrt(t_1) * -Math.sqrt((2.0 * C))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (t_0 * F)
	tmp = 0
	if C <= -5.8e-180:
		tmp = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0
	elif C <= 1.45e+87:
		tmp = -math.sqrt((t_1 * (C + math.hypot(C, B)))) / t_0
	else:
		tmp = (math.sqrt(t_1) * -math.sqrt((2.0 * C))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	tmp = 0.0
	if (C <= -5.8e-180)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / t_0);
	elseif (C <= 1.45e+87)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(C + hypot(C, B))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (t_0 * F);
	tmp = 0.0;
	if (C <= -5.8e-180)
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	elseif (C <= 1.45e+87)
		tmp = -sqrt((t_1 * (C + hypot(C, B)))) / t_0;
	else
		tmp = (sqrt(t_1) * -sqrt((2.0 * C))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -5.8e-180], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 1.45e+87], N[((-N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -5.79999999999999961e-180

   1. Initial program 18.4%

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

      1. associate-*l* 18.4%

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

      2. unpow2 18.4%

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

      3. +-commutative 18.4%

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

      4. unpow2 18.4%

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

      5. associate-*l* 18.4%

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

      6. unpow2 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in A around -inf 7.8%

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

   5. Taylor expanded in F around 0 7.8%

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

      1. cancel-sign-sub-inv 7.8%

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

      2. metadata-eval 7.8%

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

      3. unpow2 7.8%

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

      4. associate-*r* 7.8%

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

   7. Simplified 7.8%

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

   if -5.79999999999999961e-180 < C < 1.4499999999999999e87

   1. Initial program 35.5%

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

      1. associate-*l* 35.5%

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

      2. unpow2 35.5%

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

      3. +-commutative 35.5%

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

      4. unpow2 35.5%

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

      5. associate-*l* 35.5%

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

      6. unpow2 35.5%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in A around 0 33.1%

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

      1. +-commutative 33.1%

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

      2. unpow2 33.1%

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

      3. unpow2 33.1%

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

      4. hypot-def 33.6%

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

   6. Simplified 33.6%

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

   if 1.4499999999999999e87 < C

   1. Initial program 6.6%

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

      1. associate-*l* 6.6%

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

      2. unpow2 6.6%

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

      3. +-commutative 6.6%

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

      4. unpow2 6.6%

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

      5. associate-*l* 6.6%

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

      6. unpow2 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 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 45.4%

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

      2. *-commutative 45.4%

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

      3. cancel-sign-sub-inv 45.4%

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

      4. metadata-eval 45.4%

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

      5. *-commutative 45.4%

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

      6. fma-def 45.4%

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

      7. associate-*r* 45.4%

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

      8. *-commutative 45.4%

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

   6. Applied egg-rr 45.4%

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

      1. fma-udef 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*r* 45.4%

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

      4. *-commutative 45.4%

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

      5. metadata-eval 45.4%

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

      6. cancel-sign-sub-inv 45.4%

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

   8. Applied egg-rr 45.4%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 8: 31.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(t_0 \cdot F\right)\\ \mathbf{if}\;C \leq -1.7 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(B + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))) (t_1 (* 2.0 (* t_0 F))))
   (if (<= C -1.7e-184)
     (/ (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0)))))))) t_0)
     (if (<= C 3.5e-172)
       (/ (- (sqrt (* t_1 (+ A (+ B C))))) t_0)
       (/ (* (sqrt t_1) (- (sqrt (* 2.0 C)))) t_0)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (t_0 * F);
	double tmp;
	if (C <= -1.7e-184) {
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (C <= 3.5e-172) {
		tmp = -sqrt((t_1 * (A + (B + C)))) / t_0;
	} else {
		tmp = (sqrt(t_1) * -sqrt((2.0 * C))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = 2.0d0 * (t_0 * f)
    if (c <= (-1.7d-184)) then
        tmp = -sqrt((4.0d0 * (c * (f * ((b * b) + (c * (a * (-4.0d0)))))))) / t_0
    else if (c <= 3.5d-172) then
        tmp = -sqrt((t_1 * (a + (b + c)))) / t_0
    else
        tmp = (sqrt(t_1) * -sqrt((2.0d0 * c))) / t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = 2.0 * (t_0 * F);
	double tmp;
	if (C <= -1.7e-184) {
		tmp = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (C <= 3.5e-172) {
		tmp = -Math.sqrt((t_1 * (A + (B + C)))) / t_0;
	} else {
		tmp = (Math.sqrt(t_1) * -Math.sqrt((2.0 * C))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = 2.0 * (t_0 * F)
	tmp = 0
	if C <= -1.7e-184:
		tmp = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0
	elif C <= 3.5e-172:
		tmp = -math.sqrt((t_1 * (A + (B + C)))) / t_0
	else:
		tmp = (math.sqrt(t_1) * -math.sqrt((2.0 * C))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * Float64(t_0 * F))
	tmp = 0.0
	if (C <= -1.7e-184)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / t_0);
	elseif (C <= 3.5e-172)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(B + C))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = 2.0 * (t_0 * F);
	tmp = 0.0;
	if (C <= -1.7e-184)
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	elseif (C <= 3.5e-172)
		tmp = -sqrt((t_1 * (A + (B + C)))) / t_0;
	else
		tmp = (sqrt(t_1) * -sqrt((2.0 * C))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.7e-184], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 3.5e-172], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.70000000000000002e-184

   1. Initial program 18.4%

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

      1. associate-*l* 18.4%

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

      2. unpow2 18.4%

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

      3. +-commutative 18.4%

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

      4. unpow2 18.4%

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

      5. associate-*l* 18.4%

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

      6. unpow2 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in A around -inf 7.8%

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

   5. Taylor expanded in F around 0 7.8%

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

      1. cancel-sign-sub-inv 7.8%

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

      2. metadata-eval 7.8%

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

      3. unpow2 7.8%

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

      4. associate-*r* 7.8%

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

   7. Simplified 7.8%

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

   if -1.70000000000000002e-184 < C < 3.50000000000000029e-172

   1. Initial program 38.7%

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

      1. associate-*l* 38.7%

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

      2. unpow2 38.7%

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

      3. +-commutative 38.7%

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

      4. unpow2 38.7%

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

      5. associate-*l* 38.7%

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

      6. unpow2 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in B around inf 21.3%

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

   if 3.50000000000000029e-172 < C

   1. Initial program 21.0%

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

      1. associate-*l* 21.0%

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

      2. unpow2 21.0%

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

      3. +-commutative 21.0%

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

      4. unpow2 21.0%

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

      5. associate-*l* 21.0%

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

      6. unpow2 21.0%

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

   3. Simplified 21.0%

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

   4. Taylor expanded in A around -inf 31.0%

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

      1. sqrt-prod 39.4%

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

      2. *-commutative 39.4%

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

      3. cancel-sign-sub-inv 39.4%

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

      4. metadata-eval 39.4%

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

      5. *-commutative 39.4%

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

      6. fma-def 39.4%

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

      7. associate-*r* 39.4%

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

      8. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. fma-udef 39.4%

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

      2. *-commutative 39.4%

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

      3. associate-*r* 39.4%

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

      4. *-commutative 39.4%

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

      5. metadata-eval 39.4%

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

      6. cancel-sign-sub-inv 39.4%

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

   8. Applied egg-rr 39.4%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.7 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-172}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 9: 28.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -1.7 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 6.6 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C))))
        (t_1
         (/ (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0)))))))) t_0)))
   (if (<= C -1.7e-182)
     t_1
     (if (<= C 3.5e-171)
       (/ (- (sqrt (* (* 2.0 (* t_0 F)) (+ A (+ B C))))) t_0)
       (if (<= C 6.6e+87)
         t_1
         (/
          (* (sqrt (* 2.0 (* F (* A (* C -4.0))))) (- (sqrt (* 2.0 C))))
          t_0))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	double tmp;
	if (C <= -1.7e-182) {
		tmp = t_1;
	} else if (C <= 3.5e-171) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else if (C <= 6.6e+87) {
		tmp = t_1;
	} else {
		tmp = (sqrt((2.0 * (F * (A * (C * -4.0))))) * -sqrt((2.0 * C))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    t_1 = -sqrt((4.0d0 * (c * (f * ((b * b) + (c * (a * (-4.0d0)))))))) / t_0
    if (c <= (-1.7d-182)) then
        tmp = t_1
    else if (c <= 3.5d-171) then
        tmp = -sqrt(((2.0d0 * (t_0 * f)) * (a + (b + c)))) / t_0
    else if (c <= 6.6d+87) then
        tmp = t_1
    else
        tmp = (sqrt((2.0d0 * (f * (a * (c * (-4.0d0)))))) * -sqrt((2.0d0 * c))) / t_0
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double t_1 = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	double tmp;
	if (C <= -1.7e-182) {
		tmp = t_1;
	} else if (C <= 3.5e-171) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else if (C <= 6.6e+87) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt((2.0 * (F * (A * (C * -4.0))))) * -Math.sqrt((2.0 * C))) / t_0;
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	t_1 = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0
	tmp = 0
	if C <= -1.7e-182:
		tmp = t_1
	elif C <= 3.5e-171:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0
	elif C <= 6.6e+87:
		tmp = t_1
	else:
		tmp = (math.sqrt((2.0 * (F * (A * (C * -4.0))))) * -math.sqrt((2.0 * C))) / t_0
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / t_0)
	tmp = 0.0
	if (C <= -1.7e-182)
		tmp = t_1;
	elseif (C <= 3.5e-171)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(A + Float64(B + C))))) / t_0);
	elseif (C <= 6.6e+87)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(A * Float64(C * -4.0))))) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	t_1 = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	tmp = 0.0;
	if (C <= -1.7e-182)
		tmp = t_1;
	elseif (C <= 3.5e-171)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	elseif (C <= 6.6e+87)
		tmp = t_1;
	else
		tmp = (sqrt((2.0 * (F * (A * (C * -4.0))))) * -sqrt((2.0 * C))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, If[LessEqual[C, -1.7e-182], t$95$1, If[LessEqual[C, 3.5e-171], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 6.6e+87], t$95$1, N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.69999999999999995e-182 or 3.49999999999999994e-171 < C < 6.6000000000000003e87

   1. Initial program 23.8%

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

      1. associate-*l* 23.8%

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

      2. unpow2 23.8%

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

      3. +-commutative 23.8%

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

      4. unpow2 23.8%

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

      5. associate-*l* 23.8%

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

      6. unpow2 23.8%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in A around -inf 18.5%

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

   5. Taylor expanded in F around 0 18.5%

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

      1. cancel-sign-sub-inv 18.5%

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

      2. metadata-eval 18.5%

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

      3. unpow2 18.5%

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

      4. associate-*r* 18.5%

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

   7. Simplified 18.5%

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

   if -1.69999999999999995e-182 < C < 3.49999999999999994e-171

   1. Initial program 38.7%

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

      1. associate-*l* 38.7%

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

      2. unpow2 38.7%

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

      3. +-commutative 38.7%

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

      4. unpow2 38.7%

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

      5. associate-*l* 38.7%

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

      6. unpow2 38.7%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in B around inf 21.3%

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

   if 6.6000000000000003e87 < C

   1. Initial program 6.6%

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

      1. associate-*l* 6.6%

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

      2. unpow2 6.6%

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

      3. +-commutative 6.6%

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

      4. unpow2 6.6%

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

      5. associate-*l* 6.6%

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

      6. unpow2 6.6%

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

   3. Simplified 6.6%

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

   4. Taylor expanded in A around -inf 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 45.4%

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

      2. *-commutative 45.4%

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

      3. cancel-sign-sub-inv 45.4%

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

      4. metadata-eval 45.4%

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

      5. *-commutative 45.4%

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

      6. fma-def 45.4%

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

      7. associate-*r* 45.4%

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

      8. *-commutative 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in B around 0 40.9%

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

      1. *-commutative 40.9%

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

      2. associate-*r* 40.9%

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

   9. Simplified 40.9%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.7 \cdot 10^{-182}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;C \leq 6.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 10: 30.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 13600:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.42 \cdot 10^{+107}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sqrt{F} \cdot \sqrt{C}\right) \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 13600.0)
     (/ (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0)))))))) t_0)
     (if (<= B 1.42e+107)
       (/ (- (sqrt (* (* 2.0 (* t_0 F)) (+ A (+ B C))))) t_0)
       (* -2.0 (* (* (sqrt F) (sqrt C)) (/ 1.0 B)))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 13600.0) {
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (B <= 1.42e+107) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else {
		tmp = -2.0 * ((sqrt(F) * sqrt(C)) * (1.0 / B));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 13600.0d0) then
        tmp = -sqrt((4.0d0 * (c * (f * ((b * b) + (c * (a * (-4.0d0)))))))) / t_0
    else if (b <= 1.42d+107) then
        tmp = -sqrt(((2.0d0 * (t_0 * f)) * (a + (b + c)))) / t_0
    else
        tmp = (-2.0d0) * ((sqrt(f) * sqrt(c)) * (1.0d0 / b))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 13600.0) {
		tmp = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (B <= 1.42e+107) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else {
		tmp = -2.0 * ((Math.sqrt(F) * Math.sqrt(C)) * (1.0 / B));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 13600.0:
		tmp = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0
	elif B <= 1.42e+107:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0
	else:
		tmp = -2.0 * ((math.sqrt(F) * math.sqrt(C)) * (1.0 / B))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 13600.0)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / t_0);
	elseif (B <= 1.42e+107)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(A + Float64(B + C))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64(Float64(sqrt(F) * sqrt(C)) * Float64(1.0 / B)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 13600.0)
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	elseif (B <= 1.42e+107)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	else
		tmp = -2.0 * ((sqrt(F) * sqrt(C)) * (1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 13600.0], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.42e+107], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 13600

   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 19.6%

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

   5. Taylor expanded in F around 0 19.6%

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

      1. cancel-sign-sub-inv 19.6%

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

      2. metadata-eval 19.6%

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

      3. unpow2 19.6%

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

      4. associate-*r* 19.6%

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

   7. Simplified 19.6%

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

   if 13600 < B < 1.42000000000000006e107

   1. Initial program 57.5%

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

      1. associate-*l* 57.5%

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

      2. unpow2 57.5%

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

      3. +-commutative 57.5%

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

      4. unpow2 57.5%

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

      5. associate-*l* 57.5%

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

      6. unpow2 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in B around inf 52.7%

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

   if 1.42000000000000006e107 < B

   1. Initial program 0.9%

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

      1. associate-*l* 0.9%

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

      2. unpow2 0.9%

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

      3. +-commutative 0.9%

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

      4. unpow2 0.9%

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

      5. associate-*l* 0.9%

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

      6. unpow2 0.9%

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

   3. Simplified 0.9%

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

   4. Taylor expanded in A around -inf 0.6%

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

   5. Taylor expanded in B around inf 5.9%

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

      1. sqrt-prod 13.2%

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

   7. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

   9. Simplified 13.2%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 13600:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.42 \cdot 10^{+107}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(\sqrt{F} \cdot \sqrt{C}\right) \cdot \frac{1}{B}\right)\\ \end{array} \]

Alternative 11: 29.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* A C)))))
   (if (<= B 3.1)
     (/ (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0)))))))) t_0)
     (if (<= B 3.9e+108)
       (/ (- (sqrt (* (* 2.0 (* t_0 F)) (+ A (+ B C))))) t_0)
       (* -2.0 (/ (sqrt (* C F)) B))))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.1) {
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (B <= 3.9e+108) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (a * c))
    if (b <= 3.1d0) then
        tmp = -sqrt((4.0d0 * (c * (f * ((b * b) + (c * (a * (-4.0d0)))))))) / t_0
    else if (b <= 3.9d+108) then
        tmp = -sqrt(((2.0d0 * (t_0 * f)) * (a + (b + c)))) / t_0
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (A * C));
	double tmp;
	if (B <= 3.1) {
		tmp = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	} else if (B <= 3.9e+108) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (A * C))
	tmp = 0
	if B <= 3.1:
		tmp = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0
	elif B <= 3.9e+108:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (B <= 3.1)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / t_0);
	elseif (B <= 3.9e+108)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(A + Float64(B + C))))) / t_0);
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (A * C));
	tmp = 0.0;
	if (B <= 3.1)
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / t_0;
	elseif (B <= 3.9e+108)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (B + C)))) / t_0;
	else
		tmp = -2.0 * (sqrt((C * F)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.1], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 3.9e+108], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(B + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.10000000000000009

   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 19.6%

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

   5. Taylor expanded in F around 0 19.6%

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

      1. cancel-sign-sub-inv 19.6%

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

      2. metadata-eval 19.6%

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

      3. unpow2 19.6%

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

      4. associate-*r* 19.6%

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

   7. Simplified 19.6%

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

   if 3.10000000000000009 < B < 3.89999999999999985e108

   1. Initial program 57.5%

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

      1. associate-*l* 57.5%

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

      2. unpow2 57.5%

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

      3. +-commutative 57.5%

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

      4. unpow2 57.5%

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

      5. associate-*l* 57.5%

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

      6. unpow2 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in B around inf 52.7%

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

   if 3.89999999999999985e108 < B

   1. Initial program 0.9%

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

      1. associate-*l* 0.9%

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

      2. unpow2 0.9%

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

      3. +-commutative 0.9%

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

      4. unpow2 0.9%

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

      5. associate-*l* 0.9%

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

      6. unpow2 0.9%

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

   3. Simplified 0.9%

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

   4. Taylor expanded in A around -inf 0.6%

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

   5. Taylor expanded in B around inf 1.6%

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

   6. Taylor expanded in A around 0 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-commutative 5.9%

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

      3. *-rgt-identity 5.9%

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

   8. Simplified 5.9%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(B + C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 12: 28.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 1.1e+52)
   (/
    (- (sqrt (* 4.0 (* C (* F (+ (* B B) (* C (* A -4.0))))))))
    (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (/ (sqrt (* C F)) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.1e+52) {
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 1.1d+52) then
        tmp = -sqrt((4.0d0 * (c * (f * ((b * b) + (c * (a * (-4.0d0)))))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b)
    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 <= 1.1e+52) {
		tmp = -Math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 1.1e+52:
		tmp = -math.sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 1.1e+52)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64(Float64(B * B) + Float64(C * Float64(A * -4.0)))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 1.1e+52)
		tmp = -sqrt((4.0 * (C * (F * ((B * B) + (C * (A * -4.0))))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * (sqrt((C * F)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 1.1e+52], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[(B * B), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.1e52

   1. Initial program 27.6%

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

      1. associate-*l* 27.6%

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

      2. unpow2 27.6%

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

      3. +-commutative 27.6%

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

      4. unpow2 27.6%

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

      5. associate-*l* 27.6%

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

      6. unpow2 27.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in A around -inf 18.9%

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

   5. Taylor expanded in F around 0 18.9%

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

      1. cancel-sign-sub-inv 18.9%

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

      2. metadata-eval 18.9%

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

      3. unpow2 18.9%

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

      4. associate-*r* 18.9%

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

   7. Simplified 18.9%

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

   if 1.1e52 < B

   1. Initial program 12.9%

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

      1. associate-*l* 12.9%

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

      2. unpow2 12.9%

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

      3. +-commutative 12.9%

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

      4. unpow2 12.9%

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

      5. associate-*l* 12.9%

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

      6. unpow2 12.9%

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

   3. Simplified 12.9%

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

   4. Taylor expanded in A around -inf 3.2%

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

   5. Taylor expanded in B around inf 1.8%

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

   6. Taylor expanded in A around 0 7.3%

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

      1. associate-*r/ 7.3%

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

      2. *-commutative 7.3%

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

      3. *-rgt-identity 7.3%

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

   8. Simplified 7.3%

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

4. Final simplification 16.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 13: 18.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{+34}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 1.45e+34)
   (/ (- (sqrt (* -16.0 (* A (* F (* C C)))))) (- (* B B) (* 4.0 (* A C))))
   (* -2.0 (/ (sqrt (* C F)) B))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 1.45e+34) {
		tmp = -sqrt((-16.0 * (A * (F * (C * C))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 1.45d+34) then
        tmp = -sqrt(((-16.0d0) * (a * (f * (c * c))))) / ((b * b) - (4.0d0 * (a * c)))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b)
    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 <= 1.45e+34) {
		tmp = -Math.sqrt((-16.0 * (A * (F * (C * C))))) / ((B * B) - (4.0 * (A * C)));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 1.45e+34:
		tmp = -math.sqrt((-16.0 * (A * (F * (C * C))))) / ((B * B) - (4.0 * (A * C)))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 1.45e+34)
		tmp = Float64(Float64(-sqrt(Float64(-16.0 * Float64(A * Float64(F * Float64(C * C)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 1.45e+34)
		tmp = -sqrt((-16.0 * (A * (F * (C * C))))) / ((B * B) - (4.0 * (A * C)));
	else
		tmp = -2.0 * (sqrt((C * F)) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 1.45e+34], N[((-N[Sqrt[N[(-16.0 * N[(A * N[(F * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.4500000000000001e34

   1. Initial program 26.2%

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

      1. associate-*l* 26.2%

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

      2. unpow2 26.2%

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

      3. +-commutative 26.2%

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

      4. unpow2 26.2%

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

      5. associate-*l* 26.2%

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

      6. unpow2 26.2%

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

   3. Simplified 26.2%

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

   4. Taylor expanded in A around -inf 19.6%

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

   5. Taylor expanded in B around 0 11.9%

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

      1. unpow2 11.9%

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

   7. Simplified 11.9%

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

   if 1.4500000000000001e34 < B

   1. Initial program 19.9%

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

      1. associate-*l* 19.9%

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

      2. unpow2 19.9%

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

      3. +-commutative 19.9%

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

      4. unpow2 19.9%

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

      5. associate-*l* 19.9%

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

      6. unpow2 19.9%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in A around -inf 3.0%

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

   5. Taylor expanded in B around inf 1.9%

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

   6. Taylor expanded in A around 0 6.7%

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

      1. associate-*r/ 6.7%

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

      2. *-commutative 6.7%

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

      3. *-rgt-identity 6.7%

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

   8. Simplified 6.7%

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

4. Final simplification 10.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.45 \cdot 10^{+34}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 14: 9.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{B} \cdot {\left(C \cdot F\right)}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 6.2e-305)
   (* (sqrt (* C F)) (/ 2.0 B))
   (* -2.0 (* (/ 1.0 B) (pow (* C F) 0.5)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 6.2e-305) {
		tmp = sqrt((C * F)) * (2.0 / B);
	} else {
		tmp = -2.0 * ((1.0 / B) * pow((C * F), 0.5));
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b <= 6.2d-305) then
        tmp = sqrt((c * f)) * (2.0d0 / b)
    else
        tmp = (-2.0d0) * ((1.0d0 / b) * ((c * f) ** 0.5d0))
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 6.2e-305) {
		tmp = Math.sqrt((C * F)) * (2.0 / B);
	} else {
		tmp = -2.0 * ((1.0 / B) * Math.pow((C * F), 0.5));
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if B <= 6.2e-305:
		tmp = math.sqrt((C * F)) * (2.0 / B)
	else:
		tmp = -2.0 * ((1.0 / B) * math.pow((C * F), 0.5))
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 6.2e-305)
		tmp = Float64(sqrt(Float64(C * F)) * Float64(2.0 / B));
	else
		tmp = Float64(-2.0 * Float64(Float64(1.0 / B) * (Float64(C * F) ^ 0.5)));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (B <= 6.2e-305)
		tmp = sqrt((C * F)) * (2.0 / B);
	else
		tmp = -2.0 * ((1.0 / B) * ((C * F) ^ 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 6.2e-305], N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(2.0 / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(1.0 / B), $MachinePrecision] * N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.1999999999999997e-305

   1. Initial program 23.9%

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

      1. associate-*l* 23.9%

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

      2. unpow2 23.9%

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

      3. +-commutative 23.9%

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

      4. unpow2 23.9%

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

      5. associate-*l* 23.9%

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

      6. unpow2 23.9%

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

   3. Simplified 23.9%

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

   4. Taylor expanded in A around -inf 16.1%

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

      1. sqrt-prod 18.1%

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

      2. *-commutative 18.1%

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

      3. cancel-sign-sub-inv 18.1%

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

      4. metadata-eval 18.1%

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

      5. *-commutative 18.1%

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

      6. fma-def 18.1%

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

      7. associate-*r* 18.1%

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

      8. *-commutative 18.1%

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

   6. Applied egg-rr 18.1%

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

   7. Taylor expanded in B around -inf 5.8%

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

      1. unpow2 5.8%

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

      2. rem-square-sqrt 5.9%

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

   9. Simplified 5.9%

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

   if 6.1999999999999997e-305 < B

   1. Initial program 25.4%

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

      1. associate-*l* 25.4%

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

      2. unpow2 25.4%

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

      3. +-commutative 25.4%

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

      4. unpow2 25.4%

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

      5. associate-*l* 25.4%

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

      6. unpow2 25.4%

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

   3. Simplified 25.4%

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

   4. Taylor expanded in A around -inf 15.7%

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

   5. Taylor expanded in B around inf 5.5%

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

      1. pow1/2 5.6%

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

      2. *-commutative 5.6%

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

   7. Applied egg-rr 5.6%

      \[\leadsto -2 \cdot \left(\color{blue}{{\left(F \cdot C\right)}^{0.5}} \cdot \frac{1}{B}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{B} \cdot {\left(C \cdot F\right)}^{0.5}\right)\\ \end{array} \]

Alternative 15: 9.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{C \cdot F}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0 \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* C F))))
   (if (<= B -5e-310) (* t_0 (/ 2.0 B)) (* -2.0 (/ t_0 B)))))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((C * F));
	double tmp;
	if (B <= -5e-310) {
		tmp = t_0 * (2.0 / B);
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((c * f))
    if (b <= (-5d-310)) then
        tmp = t_0 * (2.0d0 / b)
    else
        tmp = (-2.0d0) * (t_0 / b)
    end if
    code = tmp
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((C * F));
	double tmp;
	if (B <= -5e-310) {
		tmp = t_0 * (2.0 / B);
	} else {
		tmp = -2.0 * (t_0 / B);
	}
	return tmp;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	t_0 = math.sqrt((C * F))
	tmp = 0
	if B <= -5e-310:
		tmp = t_0 * (2.0 / B)
	else:
		tmp = -2.0 * (t_0 / B)
	return tmp

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(C * F))
	tmp = 0.0
	if (B <= -5e-310)
		tmp = Float64(t_0 * Float64(2.0 / B));
	else
		tmp = Float64(-2.0 * Float64(t_0 / B));
	end
	return tmp
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((C * F));
	tmp = 0.0;
	if (B <= -5e-310)
		tmp = t_0 * (2.0 / B);
	else
		tmp = -2.0 * (t_0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -5e-310], N[(t$95$0 * N[(2.0 / B), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -4.999999999999985e-310

   1. Initial program 24.1%

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

      1. associate-*l* 24.1%

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

      2. unpow2 24.1%

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

      3. +-commutative 24.1%

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

      4. unpow2 24.1%

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

      5. associate-*l* 24.1%

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

      6. unpow2 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in A around -inf 16.2%

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

      1. sqrt-prod 18.2%

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

      2. *-commutative 18.2%

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

      3. cancel-sign-sub-inv 18.2%

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

      4. metadata-eval 18.2%

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

      5. *-commutative 18.2%

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

      6. fma-def 18.2%

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

      7. associate-*r* 18.2%

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

      8. *-commutative 18.2%

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

   6. Applied egg-rr 18.2%

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

   7. Taylor expanded in B around -inf 5.9%

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

      1. unpow2 5.9%

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

      2. rem-square-sqrt 5.9%

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

   9. Simplified 5.9%

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

   if -4.999999999999985e-310 < B

   1. Initial program 25.3%

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

      1. associate-*l* 25.3%

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

      2. unpow2 25.3%

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

      3. +-commutative 25.3%

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

      4. unpow2 25.3%

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

      5. associate-*l* 25.3%

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

      6. unpow2 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in A around -inf 15.6%

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

   5. Taylor expanded in B around inf 3.5%

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

   6. Taylor expanded in A around 0 5.5%

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

      1. associate-*r/ 5.5%

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

      2. *-commutative 5.5%

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

      3. *-rgt-identity 5.5%

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

   8. Simplified 5.5%

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

4. Final simplification 5.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]

Alternative 16: 5.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ -2 \cdot \frac{\sqrt{C \cdot F}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* C F)) B)))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((C * F)) / B);
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (sqrt((c * f)) / b)
end function

Java

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

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((C * F)) / B)

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B))
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((C * F)) / B);
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

   1. associate-*l* 24.8%

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

   2. unpow2 24.8%

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

   3. +-commutative 24.8%

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

   4. unpow2 24.8%

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

   5. associate-*l* 24.8%

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

   6. unpow2 24.8%

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

3. Simplified 24.8%

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

4. Taylor expanded in A around -inf 15.9%

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

5. Taylor expanded in B around inf 2.5%

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

6. Taylor expanded in A around 0 3.7%

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

   1. associate-*r/ 3.7%

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

   2. *-commutative 3.7%

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

   3. *-rgt-identity 3.7%

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

8. Simplified 3.7%

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

9. Final simplification 3.7%

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

Alternative 17: 3.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} [A, C] = \mathsf{sort}([A, C])\\ \\ \frac{-\sqrt{C \cdot F}}{B} \end{array} \]

FPCore

NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (/ (- (sqrt (* C F))) B))

C

assert(A < C);
double code(double A, double B, double C, double F) {
	return -sqrt((C * F)) / B;
}

Fortran

NOTE: A and C should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((c * f)) / b
end function

Java

assert A < C;
public static double code(double A, double B, double C, double F) {
	return -Math.sqrt((C * F)) / B;
}

Python

[A, C] = sort([A, C])
def code(A, B, C, F):
	return -math.sqrt((C * F)) / B

Julia

A, C = sort([A, C])
function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(C * F))) / B)
end

MATLAB

A, C = num2cell(sort([A, C])){:}
function tmp = code(A, B, C, F)
	tmp = -sqrt((C * F)) / B;
end

Wolfram

NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

   1. associate-*l* 24.8%

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

   2. unpow2 24.8%

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

   3. +-commutative 24.8%

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

   4. unpow2 24.8%

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

   5. associate-*l* 24.8%

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

   6. unpow2 24.8%

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

3. Simplified 24.8%

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

4. Taylor expanded in A around -inf 15.9%

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

5. Taylor expanded in B around inf 2.5%

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

6. Taylor expanded in A around inf 2.3%

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

   1. mul-1-neg 2.3%

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

   2. associate-*r/ 2.3%

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

   3. *-commutative 2.3%

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

   4. *-rgt-identity 2.3%

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

   5. distribute-neg-frac 2.3%

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

8. Simplified 2.3%

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

9. Final simplification 2.3%

   \[\leadsto \frac{-\sqrt{C \cdot F}}{B} \]

Reproduce

herbie shell --seed 2023196 
(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))))