ABCF->ab-angle a

Percentage Accurate: 18.8% → 52.0%

Time: 33.0s

Alternatives: 20

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= (pow B 2.0) 5e+156)
     (/
      (* (sqrt (+ C (+ A (hypot B (- A C))))) (- (sqrt (* 2.0 (* F t_0)))))
      t_0)
     (* (* (sqrt (+ A (hypot B A))) (sqrt F)) (/ (- (sqrt 2.0)) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (pow(B, 2.0) <= 5e+156) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * t_0)))) / t_0;
	} else {
		tmp = (sqrt((A + hypot(B, A))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (Math.pow(B, 2.0) <= 5e+156) {
		tmp = (Math.sqrt((C + (A + Math.hypot(B, (A - C))))) * -Math.sqrt((2.0 * (F * t_0)))) / t_0;
	} else {
		tmp = (Math.sqrt((A + Math.hypot(B, A))) * Math.sqrt(F)) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if math.pow(B, 2.0) <= 5e+156:
		tmp = (math.sqrt((C + (A + math.hypot(B, (A - C))))) * -math.sqrt((2.0 * (F * t_0)))) / t_0
	else:
		tmp = (math.sqrt((A + math.hypot(B, A))) * math.sqrt(F)) * (-math.sqrt(2.0) / B)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e+156)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_0))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B, A))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if ((B ^ 2.0) <= 5e+156)
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * t_0)))) / t_0;
	else
		tmp = (sqrt((A + hypot(B, A))) * sqrt(F)) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e+156], 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 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 4.99999999999999992e156

   1. Initial program 27.2%

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

      1. associate-*l* 27.2%

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

      2. unpow2 27.2%

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

      3. +-commutative 27.2%

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

      4. unpow2 27.2%

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

      5. associate-*l* 27.2%

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

      6. unpow2 27.2%

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

   3. Simplified 27.2%

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

      1. sqrt-prod 30.1%

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

      2. *-commutative 30.1%

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

      3. *-commutative 30.1%

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

      4. associate-+l+ 30.5%

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

      5. unpow2 30.5%

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

      6. hypot-udef 41.6%

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

      7. associate-+r+ 40.5%

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

      8. +-commutative 40.5%

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

      9. associate-+r+ 41.2%

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

   5. Applied egg-rr 41.2%

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

   if 4.99999999999999992e156 < (pow.f64 B 2)

   1. Initial program 6.0%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in C around 0 10.0%

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

      1. mul-1-neg 10.0%

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

      2. unpow2 10.0%

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

      3. unpow2 10.0%

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

   6. Simplified 10.0%

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

      1. sqrt-prod 10.0%

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

      2. hypot-def 36.4%

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

   8. Applied egg-rr 36.4%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq 9.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(2 \cdot A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+217} \lor \neg \left(B \leq 2.6 \cdot 10^{+268}\right):\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A))))
        (t_1
         (* (sqrt (+ C (+ A (hypot B (- A C))))) (- (sqrt (* 2.0 (* F t_0))))))
        (t_2 (fma C (* A -4.0) (* B B))))
   (if (<= B 9.6e-224)
     (/ t_1 (* (* C A) -4.0))
     (if (<= B 6.1e-176)
       (/ (- (sqrt (* 2.0 (* (* F t_2) (* 2.0 A))))) t_2)
       (if (<= B 2.6e+81)
         (/ t_1 t_0)
         (if (or (<= B 3.2e+217) (not (<= B 2.6e+268)))
           (* (sqrt (* F (+ A (hypot B A)))) (/ (- (sqrt 2.0)) B))
           (* (sqrt 2.0) (- (sqrt (/ F B))))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * t_0)));
	double t_2 = fma(C, (A * -4.0), (B * B));
	double tmp;
	if (B <= 9.6e-224) {
		tmp = t_1 / ((C * A) * -4.0);
	} else if (B <= 6.1e-176) {
		tmp = -sqrt((2.0 * ((F * t_2) * (2.0 * A)))) / t_2;
	} else if (B <= 2.6e+81) {
		tmp = t_1 / t_0;
	} else if ((B <= 3.2e+217) || !(B <= 2.6e+268)) {
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_0)))))
	t_2 = fma(C, Float64(A * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= 9.6e-224)
		tmp = Float64(t_1 / Float64(Float64(C * A) * -4.0));
	elseif (B <= 6.1e-176)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_2) * Float64(2.0 * A))))) / t_2);
	elseif (B <= 2.6e+81)
		tmp = Float64(t_1 / t_0);
	elseif ((B <= 3.2e+217) || !(B <= 2.6e+268))
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B, A)))) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9.6e-224], N[(t$95$1 / N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 6.1e-176], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$2), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 2.6e+81], N[(t$95$1 / t$95$0), $MachinePrecision], If[Or[LessEqual[B, 3.2e+217], N[Not[LessEqual[B, 2.6e+268]], $MachinePrecision]], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 9.60000000000000057e-224

   1. Initial program 24.4%

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

      1. associate-*l* 24.4%

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

      2. unpow2 24.4%

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

      3. +-commutative 24.4%

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

      4. unpow2 24.4%

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

      5. associate-*l* 24.4%

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

      6. unpow2 24.4%

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

   3. Simplified 24.4%

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

      1. sqrt-prod 26.4%

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

      2. *-commutative 26.4%

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

      3. *-commutative 26.4%

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

      4. associate-+l+ 26.8%

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

      5. unpow2 26.8%

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

      6. hypot-udef 34.1%

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

      7. associate-+r+ 33.2%

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

      8. +-commutative 33.2%

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

      9. associate-+r+ 33.5%

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

   5. Applied egg-rr 33.5%

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

   6. Taylor expanded in B around 0 21.1%

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

   if 9.60000000000000057e-224 < B < 6.1000000000000003e-176

   1. Initial program 14.5%

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

   3. Simplified 17.3%

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

   4. Taylor expanded in C around -inf 38.2%

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

   if 6.1000000000000003e-176 < B < 2.59999999999999992e81

   1. Initial program 17.6%

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

      1. associate-*l* 17.6%

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

      2. unpow2 17.6%

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

      3. +-commutative 17.6%

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

      4. unpow2 17.6%

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

      5. associate-*l* 17.6%

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

      6. unpow2 17.6%

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

   3. Simplified 17.6%

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

      1. sqrt-prod 21.1%

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

      2. *-commutative 21.1%

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

      3. *-commutative 21.1%

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

      4. associate-+l+ 21.1%

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

      5. unpow2 21.1%

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

      6. hypot-udef 38.4%

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

      7. associate-+r+ 38.2%

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

      8. +-commutative 38.2%

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

      9. associate-+r+ 39.3%

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

   5. Applied egg-rr 39.3%

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

   if 2.59999999999999992e81 < B < 3.2000000000000001e217 or 2.59999999999999993e268 < B

   1. Initial program 8.5%

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

   3. Simplified 9.0%

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

   4. Taylor expanded in C around 0 21.9%

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

      1. mul-1-neg 21.9%

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

      2. distribute-rgt-neg-in 21.9%

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

      3. unpow2 21.9%

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

      4. unpow2 21.9%

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

      5. hypot-def 48.4%

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

   6. Simplified 48.4%

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

   if 3.2000000000000001e217 < B < 2.59999999999999993e268

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. unpow2 0.0%

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

      6. hypot-def 2.0%

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

   6. Simplified 2.0%

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

   7. Taylor expanded in C around 0 53.9%

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

      1. mul-1-neg 53.9%

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

   9. Simplified 53.9%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}\right)}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+217} \lor \neg \left(B \leq 2.6 \cdot 10^{+268}\right):\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 3: 41.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-310)
   (-
    (/
     (*
      (sqrt (+ C (+ A (hypot B (- A C)))))
      (sqrt (* 2.0 (* -4.0 (* A (* F C))))))
     (- (* B B) (* 4.0 (* C A)))))
   (if (<= F 2e+44)
     (* (sqrt (* F (+ A (hypot B A)))) (/ (- (sqrt 2.0)) B))
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -((sqrt((C + (A + hypot(B, (A - C))))) * sqrt((2.0 * (-4.0 * (A * (F * C)))))) / ((B * B) - (4.0 * (C * A))));
	} else if (F <= 2e+44) {
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = -((Math.sqrt((C + (A + Math.hypot(B, (A - C))))) * Math.sqrt((2.0 * (-4.0 * (A * (F * C)))))) / ((B * B) - (4.0 * (C * A))));
	} else if (F <= 2e+44) {
		tmp = Math.sqrt((F * (A + Math.hypot(B, A)))) * (-Math.sqrt(2.0) / B);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = -((math.sqrt((C + (A + math.hypot(B, (A - C))))) * math.sqrt((2.0 * (-4.0 * (A * (F * C)))))) / ((B * B) - (4.0 * (C * A))))
	elif F <= 2e+44:
		tmp = math.sqrt((F * (A + math.hypot(B, A)))) * (-math.sqrt(2.0) / B)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(-Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B, Float64(A - C))))) * sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(F * C)))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))));
	elseif (F <= 2e+44)
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B, A)))) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = -((sqrt((C + (A + hypot(B, (A - C))))) * sqrt((2.0 * (-4.0 * (A * (F * C)))))) / ((B * B) - (4.0 * (C * A))));
	elseif (F <= 2e+44)
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-310], (-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[(-4.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e+44], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. unpow2 40.8%

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

      3. +-commutative 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

   3. Simplified 40.8%

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

      1. sqrt-prod 45.3%

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

      2. *-commutative 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-+l+ 45.3%

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

      5. unpow2 45.3%

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

      6. hypot-udef 59.8%

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

      7. associate-+r+ 59.8%

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

      8. +-commutative 59.8%

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

      9. associate-+r+ 59.8%

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

   5. Applied egg-rr 59.8%

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

   6. Taylor expanded in B around 0 56.8%

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

   if -4.999999999999985e-310 < F < 2.0000000000000002e44

   1. Initial program 17.4%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in C around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. distribute-rgt-neg-in 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. hypot-def 20.6%

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

   6. Simplified 20.6%

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

   if 2.0000000000000002e44 < F

   1. Initial program 14.6%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in A around 0 5.6%

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

      1. *-commutative 5.6%

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

      2. associate-*l* 5.6%

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

      3. *-commutative 5.6%

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

      4. unpow2 5.6%

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

      5. unpow2 5.6%

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

      6. hypot-def 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in C around 0 14.6%

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

      1. mul-1-neg 14.6%

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

   9. Simplified 14.6%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 4: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}\right)}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-310)
   (/
    (*
     (sqrt (+ C (+ A (hypot B (- A C)))))
     (- (sqrt (* 2.0 (* F (- (* B B) (* 4.0 (* C A))))))))
    (* (* C A) -4.0))
   (if (<= F 2.5e+44)
     (* (sqrt (* F (+ A (hypot B A)))) (/ (- (sqrt 2.0)) B))
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * ((B * B) - (4.0 * (C * A))))))) / ((C * A) * -4.0);
	} else if (F <= 2.5e+44) {
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (Math.sqrt((C + (A + Math.hypot(B, (A - C))))) * -Math.sqrt((2.0 * (F * ((B * B) - (4.0 * (C * A))))))) / ((C * A) * -4.0);
	} else if (F <= 2.5e+44) {
		tmp = Math.sqrt((F * (A + Math.hypot(B, A)))) * (-Math.sqrt(2.0) / B);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = (math.sqrt((C + (A + math.hypot(B, (A - C))))) * -math.sqrt((2.0 * (F * ((B * B) - (4.0 * (C * A))))))) / ((C * A) * -4.0)
	elif F <= 2.5e+44:
		tmp = math.sqrt((F * (A + math.hypot(B, A)))) * (-math.sqrt(2.0) / B)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		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(C * A)))))))) / Float64(Float64(C * A) * -4.0));
	elseif (F <= 2.5e+44)
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B, A)))) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = (sqrt((C + (A + hypot(B, (A - C))))) * -sqrt((2.0 * (F * ((B * B) - (4.0 * (C * A))))))) / ((C * A) * -4.0);
	elseif (F <= 2.5e+44)
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-310], 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[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e+44], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. unpow2 40.8%

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

      3. +-commutative 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

   3. Simplified 40.8%

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

      1. sqrt-prod 45.3%

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

      2. *-commutative 45.3%

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

      3. *-commutative 45.3%

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

      4. associate-+l+ 45.3%

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

      5. unpow2 45.3%

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

      6. hypot-udef 59.8%

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

      7. associate-+r+ 59.8%

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

      8. +-commutative 59.8%

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

      9. associate-+r+ 59.8%

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

   5. Applied egg-rr 59.8%

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

   6. Taylor expanded in B around 0 59.8%

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

   if -4.999999999999985e-310 < F < 2.4999999999999998e44

   1. Initial program 17.4%

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

   3. Simplified 23.8%

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

   4. Taylor expanded in C around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. distribute-rgt-neg-in 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. hypot-def 20.6%

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

   6. Simplified 20.6%

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

   if 2.4999999999999998e44 < F

   1. Initial program 14.6%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in A around 0 5.6%

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

      1. *-commutative 5.6%

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

      2. associate-*l* 5.6%

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

      3. *-commutative 5.6%

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

      4. unpow2 5.6%

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

      5. unpow2 5.6%

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

      6. hypot-def 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in C around 0 14.6%

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

      1. mul-1-neg 14.6%

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

   9. Simplified 14.6%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)}\right)}{\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 5: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 4.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 4.8e-306)
   (/
    (-
     (sqrt
      (*
       2.0
       (* F (* (+ (* B B) (* (* C A) -4.0)) (+ (hypot B (- A C)) (+ C A)))))))
    (- (* B B) (* 4.0 (* C A))))
   (if (<= F 1.16e-45)
     (/ (* (sqrt (* F (+ C (hypot C B)))) (- (sqrt 2.0))) B)
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 4.8e-306) {
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 1.16e-45) {
		tmp = (sqrt((F * (C + hypot(C, B)))) * -sqrt(2.0)) / B;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 4.8e-306) {
		tmp = -Math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (Math.hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 1.16e-45) {
		tmp = (Math.sqrt((F * (C + Math.hypot(C, B)))) * -Math.sqrt(2.0)) / B;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 4.8e-306:
		tmp = -math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (math.hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)))
	elif F <= 1.16e-45:
		tmp = (math.sqrt((F * (C + math.hypot(C, B)))) * -math.sqrt(2.0)) / B
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 4.8e-306)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0)) * Float64(hypot(B, Float64(A - C)) + Float64(C + A))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (F <= 1.16e-45)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * Float64(-sqrt(2.0))) / B);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 4.8e-306)
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	elseif (F <= 1.16e-45)
		tmp = (sqrt((F * (C + hypot(C, B)))) * -sqrt(2.0)) / B;
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 4.8e-306], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.16e-45], N[(N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision] / B), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 4.7999999999999999e-306

   1. Initial program 39.8%

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

      1. associate-*l* 39.8%

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

      2. unpow2 39.8%

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

      3. +-commutative 39.8%

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

      4. unpow2 39.8%

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

      5. associate-*l* 39.8%

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

      6. unpow2 39.8%

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

   3. Simplified 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. associate-*l* 39.8%

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

      3. *-commutative 39.8%

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

      4. *-commutative 39.8%

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

      5. unpow2 39.8%

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

      6. hypot-udef 48.2%

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

      7. +-commutative 48.2%

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

      8. associate-+r+ 48.2%

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

   5. Applied egg-rr 48.2%

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

      1. *-lft-identity 48.2%

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

      2. associate-*r* 48.2%

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

      3. fma-neg 48.2%

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

      4. *-commutative 48.2%

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

      5. *-commutative 48.2%

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

      6. distribute-rgt-neg-in 48.2%

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

      7. *-commutative 48.2%

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

      8. metadata-eval 48.2%

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

      9. associate-*r* 48.2%

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

      10. associate-*r* 48.2%

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

      11. associate-*l* 48.2%

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

   7. Simplified 48.2%

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

   if 4.7999999999999999e-306 < F < 1.16000000000000002e-45

   1. Initial program 20.1%

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

      1. associate-*l* 20.1%

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

      2. unpow2 20.1%

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

      3. +-commutative 20.1%

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

      4. unpow2 20.1%

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

      5. associate-*l* 20.1%

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

      6. unpow2 20.1%

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

   3. Simplified 20.1%

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

      1. sqrt-prod 21.7%

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

      2. *-commutative 21.7%

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

      3. *-commutative 21.7%

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

      4. associate-+l+ 21.9%

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

      5. unpow2 21.9%

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

      6. hypot-udef 30.3%

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

      7. associate-+r+ 29.7%

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

      8. +-commutative 29.7%

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

      9. associate-+r+ 30.1%

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

   5. Applied egg-rr 30.1%

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

   6. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. associate-*l/ 8.1%

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

      3. +-commutative 8.1%

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

      4. unpow2 8.1%

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

      5. unpow2 8.1%

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

      6. hypot-def 21.8%

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

   8. Simplified 21.8%

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

   if 1.16000000000000002e-45 < F

   1. Initial program 13.3%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in A around 0 7.1%

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

      1. *-commutative 7.1%

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

      2. associate-*l* 7.1%

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

      3. *-commutative 7.1%

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

      4. unpow2 7.1%

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

      5. unpow2 7.1%

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

      6. hypot-def 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in C around 0 15.2%

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

      1. mul-1-neg 15.2%

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

   9. Simplified 15.2%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.8 \cdot 10^{-306}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 6: 39.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 1.25e-302)
   (/
    (-
     (sqrt
      (*
       2.0
       (* F (* (+ (* B B) (* (* C A) -4.0)) (+ (hypot B (- A C)) (+ C A)))))))
    (- (* B B) (* 4.0 (* C A))))
   (if (<= F 2e+44)
     (* (sqrt (* F (+ A (hypot B A)))) (/ (- (sqrt 2.0)) B))
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.25e-302) {
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2e+44) {
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.25e-302) {
		tmp = -Math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (Math.hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2e+44) {
		tmp = Math.sqrt((F * (A + Math.hypot(B, A)))) * (-Math.sqrt(2.0) / B);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 1.25e-302:
		tmp = -math.sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (math.hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)))
	elif F <= 2e+44:
		tmp = math.sqrt((F * (A + math.hypot(B, A)))) * (-math.sqrt(2.0) / B)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 1.25e-302)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0)) * Float64(hypot(B, Float64(A - C)) + Float64(C + A))))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (F <= 2e+44)
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B, A)))) * Float64(Float64(-sqrt(2.0)) / B));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 1.25e-302)
		tmp = -sqrt((2.0 * (F * (((B * B) + ((C * A) * -4.0)) * (hypot(B, (A - C)) + (C + A)))))) / ((B * B) - (4.0 * (C * A)));
	elseif (F <= 2e+44)
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 1.25e-302], N[((-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e+44], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.25000000000000008e-302

   1. Initial program 37.7%

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

      1. associate-*l* 37.7%

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

      2. unpow2 37.7%

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

      3. +-commutative 37.7%

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

      4. unpow2 37.7%

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

      5. associate-*l* 37.7%

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

      6. unpow2 37.7%

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

   3. Simplified 37.7%

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

      1. *-un-lft-identity 37.7%

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

      2. associate-*l* 37.7%

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

      3. *-commutative 37.7%

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

      4. *-commutative 37.7%

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

      5. unpow2 37.7%

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

      6. hypot-udef 46.0%

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

      7. +-commutative 46.0%

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

      8. associate-+r+ 46.0%

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

   5. Applied egg-rr 46.0%

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

      1. *-lft-identity 46.0%

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

      2. associate-*r* 46.0%

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

      3. fma-neg 46.0%

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

      4. *-commutative 46.0%

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

      5. *-commutative 46.0%

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

      6. distribute-rgt-neg-in 46.0%

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

      7. *-commutative 46.0%

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

      8. metadata-eval 46.0%

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

      9. associate-*r* 46.0%

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

      10. associate-*r* 46.0%

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

      11. associate-*l* 46.0%

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

   7. Simplified 46.0%

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

   if 1.25000000000000008e-302 < F < 2.0000000000000002e44

   1. Initial program 17.8%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in C around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. distribute-rgt-neg-in 8.8%

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

      3. unpow2 8.8%

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

      4. unpow2 8.8%

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

      5. hypot-def 20.9%

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

   6. Simplified 20.9%

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

   if 2.0000000000000002e44 < F

   1. Initial program 14.6%

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

   3. Simplified 16.8%

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

   4. Taylor expanded in A around 0 5.6%

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

      1. *-commutative 5.6%

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

      2. associate-*l* 5.6%

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

      3. *-commutative 5.6%

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

      4. unpow2 5.6%

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

      5. unpow2 5.6%

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

      6. hypot-def 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in C around 0 14.6%

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

      1. mul-1-neg 14.6%

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

   9. Simplified 14.6%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 7: 37.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq 9.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+202}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (/ (sqrt 2.0) B)))
   (if (<= B 9.4e+80)
     (/ (- (sqrt (* 2.0 (* (* F t_0) (+ C (+ A (hypot B (- A C)))))))) t_0)
     (if (<= B 1.7e+202)
       (* t_1 (- (sqrt (* B F))))
       (if (<= B 6e+268)
         (* (sqrt 2.0) (- (sqrt (/ F B))))
         (* t_1 (- (sqrt (* F (+ B A))))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = sqrt(2.0) / B;
	double tmp;
	if (B <= 9.4e+80) {
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	} else if (B <= 1.7e+202) {
		tmp = t_1 * -sqrt((B * F));
	} else if (B <= 6e+268) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else {
		tmp = t_1 * -sqrt((F * (B + A)));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = Math.sqrt(2.0) / B;
	double tmp;
	if (B <= 9.4e+80) {
		tmp = -Math.sqrt((2.0 * ((F * t_0) * (C + (A + Math.hypot(B, (A - C))))))) / t_0;
	} else if (B <= 1.7e+202) {
		tmp = t_1 * -Math.sqrt((B * F));
	} else if (B <= 6e+268) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else {
		tmp = t_1 * -Math.sqrt((F * (B + A)));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = math.sqrt(2.0) / B
	tmp = 0
	if B <= 9.4e+80:
		tmp = -math.sqrt((2.0 * ((F * t_0) * (C + (A + math.hypot(B, (A - C))))))) / t_0
	elif B <= 1.7e+202:
		tmp = t_1 * -math.sqrt((B * F))
	elif B <= 6e+268:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	else:
		tmp = t_1 * -math.sqrt((F * (B + A)))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(sqrt(2.0) / B)
	tmp = 0.0
	if (B <= 9.4e+80)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_0) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_0);
	elseif (B <= 1.7e+202)
		tmp = Float64(t_1 * Float64(-sqrt(Float64(B * F))));
	elseif (B <= 6e+268)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	else
		tmp = Float64(t_1 * Float64(-sqrt(Float64(F * Float64(B + A)))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = sqrt(2.0) / B;
	tmp = 0.0;
	if (B <= 9.4e+80)
		tmp = -sqrt((2.0 * ((F * t_0) * (C + (A + hypot(B, (A - C))))))) / t_0;
	elseif (B <= 1.7e+202)
		tmp = t_1 * -sqrt((B * F));
	elseif (B <= 6e+268)
		tmp = sqrt(2.0) * -sqrt((F / B));
	else
		tmp = t_1 * -sqrt((F * (B + A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, 9.4e+80], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$0), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.7e+202], N[(t$95$1 * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 6e+268], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$1 * (-N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9.40000000000000019e80

   1. Initial program 22.4%

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

      1. associate-*l* 22.4%

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

      2. unpow2 22.4%

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

      3. +-commutative 22.4%

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

      4. unpow2 22.4%

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

      5. associate-*l* 22.4%

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

      6. unpow2 22.4%

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

   3. Simplified 22.4%

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

      1. distribute-frac-neg 22.4%

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

   5. Applied egg-rr 28.6%

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

   if 9.40000000000000019e80 < B < 1.7e202

   1. Initial program 12.6%

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

   3. Simplified 13.4%

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

   4. Taylor expanded in C around 0 31.4%

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

      1. mul-1-neg 31.4%

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

      2. unpow2 31.4%

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

      3. unpow2 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in A around 0 43.6%

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

   if 1.7e202 < B < 5.99999999999999984e268

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in A around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*l* 0.0%

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

      3. *-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. unpow2 0.0%

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

      6. hypot-def 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in C around 0 50.3%

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

      1. mul-1-neg 50.3%

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

   9. Simplified 50.3%

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

   if 5.99999999999999984e268 < B

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in C around 0 2.2%

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

      1. mul-1-neg 2.2%

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

      2. unpow2 2.2%

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

      3. unpow2 2.2%

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

   6. Simplified 2.2%

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

   7. Taylor expanded in B around inf 63.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+202}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+268}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \end{array} \]

Alternative 8: 37.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{A + \left(C + A\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-310)
   (/
    (*
     (sqrt (* 2.0 (* F (+ (* B B) (* (* C A) -4.0)))))
     (- (sqrt (+ A (+ C A)))))
    (- (* B B) (* 4.0 (* C A))))
   (if (<= F 2.25e-27)
     (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ B A)))))
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((A + (C + A)))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2.25e-27) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-5d-310)) then
        tmp = (sqrt((2.0d0 * (f * ((b * b) + ((c * a) * (-4.0d0)))))) * -sqrt((a + (c + a)))) / ((b * b) - (4.0d0 * (c * a)))
    else if (f <= 2.25d-27) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * (b + a)))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (Math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -Math.sqrt((A + (C + A)))) / ((B * B) - (4.0 * (C * A)));
	} else if (F <= 2.25e-27) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (B + A)));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = (math.sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -math.sqrt((A + (C + A)))) / ((B * B) - (4.0 * (C * A)))
	elif F <= 2.25e-27:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (B + A)))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(A + Float64(C + A))))) / Float64(Float64(B * B) - Float64(4.0 * Float64(C * A))));
	elseif (F <= 2.25e-27)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(B + A)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = (sqrt((2.0 * (F * ((B * B) + ((C * A) * -4.0))))) * -sqrt((A + (C + A)))) / ((B * B) - (4.0 * (C * A)));
	elseif (F <= 2.25e-27)
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.25e-27], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. unpow2 40.8%

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

      3. +-commutative 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in A around inf 27.1%

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

      1. sqrt-prod 36.3%

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

      2. *-commutative 36.3%

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

      3. cancel-sign-sub-inv 36.3%

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

      4. *-commutative 36.3%

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

      5. metadata-eval 36.3%

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

      6. associate-+l+ 36.3%

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

   6. Applied egg-rr 36.3%

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

   if -4.999999999999985e-310 < F < 2.2500000000000001e-27

   1. Initial program 19.7%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. unpow2 9.0%

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

      3. unpow2 9.0%

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

   6. Simplified 9.0%

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

   7. Taylor expanded in B around inf 17.2%

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

   if 2.2500000000000001e-27 < F

   1. Initial program 13.1%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in A around 0 5.8%

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

      1. *-commutative 5.8%

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

      2. associate-*l* 5.7%

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

      3. *-commutative 5.7%

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

      4. unpow2 5.7%

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

      5. unpow2 5.7%

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

      6. hypot-def 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

   9. Simplified 14.2%

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

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{A + \left(C + A\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9: 35.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= F -5e-310)
     (- (/ (sqrt (* (* 2.0 (* F t_0)) (+ (+ C A) (- C A)))) t_0))
     (if (<= F 2.1e-25)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ B A)))))
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (F <= -5e-310) {
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	} else if (F <= 2.1e-25) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (f <= (-5d-310)) then
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * ((c + a) + (c - a)))) / t_0)
    else if (f <= 2.1d-25) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * (b + a)))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (F <= -5e-310) {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	} else if (F <= 2.1e-25) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (B + A)));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if F <= -5e-310:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0)
	elif F <= 2.1e-25:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (B + A)))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(C + A) + Float64(C - A)))) / t_0));
	elseif (F <= 2.1e-25)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(B + A)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	elseif (F <= 2.1e-25)
		tmp = (sqrt(2.0) / B) * -sqrt((F * (B + A)));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e-310], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] + N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[F, 2.1e-25], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 40.8%

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

      1. associate-*l* 40.8%

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

      2. unpow2 40.8%

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

      3. +-commutative 40.8%

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

      4. unpow2 40.8%

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

      5. associate-*l* 40.8%

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

      6. unpow2 40.8%

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

   3. Simplified 40.8%

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

   4. Taylor expanded in A around -inf 28.7%

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

      1. mul-1-neg 28.7%

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

      2. sub-neg 28.7%

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

   6. Simplified 28.7%

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

   if -4.999999999999985e-310 < F < 2.10000000000000002e-25

   1. Initial program 19.7%

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

   3. Simplified 26.9%

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

   4. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. unpow2 9.0%

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

      3. unpow2 9.0%

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

   6. Simplified 9.0%

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

   7. Taylor expanded in B around inf 17.2%

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

   if 2.10000000000000002e-25 < F

   1. Initial program 13.1%

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

   3. Simplified 15.3%

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

   4. Taylor expanded in A around 0 5.8%

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

      1. *-commutative 5.8%

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

      2. associate-*l* 5.7%

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

      3. *-commutative 5.7%

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

      4. unpow2 5.7%

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

      5. unpow2 5.7%

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

      6. hypot-def 6.4%

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

   6. Simplified 6.4%

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

   7. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

   9. Simplified 14.2%

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

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10: 36.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-291}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= F -3.5e-291)
     (- (/ (sqrt (* (* 2.0 (* F t_0)) (+ (+ C A) (- C A)))) t_0))
     (if (<= F 40000000.0)
       (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
       (* (sqrt 2.0) (- (sqrt (/ F B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (F <= -3.5e-291) {
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	} else if (F <= 40000000.0) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (f <= (-3.5d-291)) then
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * ((c + a) + (c - a)))) / t_0)
    else if (f <= 40000000.0d0) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * f))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (F <= -3.5e-291) {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	} else if (F <= 40000000.0) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if F <= -3.5e-291:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0)
	elif F <= 40000000.0:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (F <= -3.5e-291)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(C + A) + Float64(C - A)))) / t_0));
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (F <= -3.5e-291)
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C - A)))) / t_0);
	elseif (F <= 40000000.0)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e-291], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] + N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[F, 40000000.0], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.49999999999999996e-291

   1. Initial program 41.4%

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

      1. associate-*l* 41.4%

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

      2. unpow2 41.4%

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

      3. +-commutative 41.4%

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

      4. unpow2 41.4%

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

      5. associate-*l* 41.4%

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

      6. unpow2 41.4%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in A around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

      2. sub-neg 31.2%

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

   6. Simplified 31.2%

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

   if -3.49999999999999996e-291 < F < 4e7

   1. Initial program 18.6%

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

   3. Simplified 26.0%

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

   4. Taylor expanded in C around 0 9.1%

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

      1. mul-1-neg 9.1%

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

      2. unpow2 9.1%

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

      3. unpow2 9.1%

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

   6. Simplified 9.1%

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

   7. Taylor expanded in A around 0 17.7%

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

   if 4e7 < F

   1. Initial program 14.1%

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

   3. Simplified 16.3%

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

   4. Taylor expanded in A around 0 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*l* 5.3%

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

      3. *-commutative 5.3%

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

      4. unpow2 5.3%

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

      5. unpow2 5.3%

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

      6. hypot-def 5.8%

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

   6. Simplified 5.8%

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

   7. Taylor expanded in C around 0 13.5%

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

      1. mul-1-neg 13.5%

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

   9. Simplified 13.5%

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

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-291}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11: 29.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -1.6 \cdot 10^{+190}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) + \left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5}{C} - A\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -1.6e+190)
     (- (/ (sqrt (* -16.0 (* (* F C) (* A A)))) t_0))
     (if (<= C 1.4e-48)
       (* (sqrt 2.0) (- (sqrt (/ F B))))
       (-
        (/
         (sqrt
          (* (* 2.0 (* F t_0)) (+ (+ C A) (+ C (- (/ (* (* B B) 0.5) C) A)))))
         t_0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -1.6e+190) {
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 1.4e-48) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else {
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (c <= (-1.6d+190)) then
        tmp = -(sqrt(((-16.0d0) * ((f * c) * (a * a)))) / t_0)
    else if (c <= 1.4d-48) then
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    else
        tmp = -(sqrt(((2.0d0 * (f * t_0)) * ((c + a) + (c + ((((b * b) * 0.5d0) / c) - a))))) / t_0)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -1.6e+190) {
		tmp = -(Math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 1.4e-48) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	} else {
		tmp = -(Math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -1.6e+190:
		tmp = -(math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0)
	elif C <= 1.4e-48:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	else:
		tmp = -(math.sqrt(((2.0 * (F * t_0)) * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -1.6e+190)
		tmp = Float64(-Float64(sqrt(Float64(-16.0 * Float64(Float64(F * C) * Float64(A * A)))) / t_0));
	elseif (C <= 1.4e-48)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(C + A) + Float64(C + Float64(Float64(Float64(Float64(B * B) * 0.5) / C) - A))))) / t_0));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -1.6e+190)
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	elseif (C <= 1.4e-48)
		tmp = sqrt(2.0) * -sqrt((F / B));
	else
		tmp = -(sqrt(((2.0 * (F * t_0)) * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.6e+190], (-N[(N[Sqrt[N[(-16.0 * N[(N[(F * C), $MachinePrecision] * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[C, 1.4e-48], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] + N[(C + N[(N[(N[(N[(B * B), $MachinePrecision] * 0.5), $MachinePrecision] / C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.6e190

   1. Initial program 0.8%

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

      1. associate-*l* 0.8%

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

      2. unpow2 0.8%

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

      3. +-commutative 0.8%

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

      4. unpow2 0.8%

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

      5. associate-*l* 0.8%

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

      6. unpow2 0.8%

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

   3. Simplified 0.8%

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

   4. Taylor expanded in A around inf 0.7%

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

   5. Taylor expanded in A around inf 23.9%

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

      1. unpow2 23.9%

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

   7. Simplified 23.9%

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

   if -1.6e190 < C < 1.40000000000000002e-48

   1. Initial program 17.8%

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

   3. Simplified 20.8%

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

   4. Taylor expanded in A around 0 6.8%

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

      1. *-commutative 6.8%

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

      2. associate-*l* 6.8%

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

      3. *-commutative 6.8%

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

      4. unpow2 6.8%

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

      5. unpow2 6.8%

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

      6. hypot-def 7.6%

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

   6. Simplified 7.6%

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

   7. Taylor expanded in C around 0 12.9%

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

      1. mul-1-neg 12.9%

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

   9. Simplified 12.9%

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

   if 1.40000000000000002e-48 < C

   1. Initial program 29.1%

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

      1. associate-*l* 29.1%

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

      2. unpow2 29.1%

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

      3. +-commutative 29.1%

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

      4. unpow2 29.1%

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

      5. associate-*l* 29.1%

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

      6. unpow2 29.1%

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

   3. Simplified 29.1%

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

   4. Taylor expanded in C around inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. unsub-neg 32.3%

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

   6. Simplified 32.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5 + 0}{C} - A\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.6 \cdot 10^{+190}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5}{C} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 12: 16.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq -9.5 \cdot 10^{-243}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(\left(C + A\right) + \left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5}{C} - A\right)\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C -9.5e-243)
     (- (/ (sqrt (* -16.0 (* (* F C) (* A A)))) t_0))
     (if (<= C 2.5e-66)
       (/ (- (sqrt (* t_1 (+ A (+ C A))))) t_0)
       (-
        (/
         (sqrt (* t_1 (+ (+ C A) (+ C (- (/ (* (* B B) 0.5) C) A)))))
         t_0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -9.5e-243) {
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 2.5e-66) {
		tmp = -sqrt((t_1 * (A + (C + A)))) / t_0;
	} else {
		tmp = -(sqrt((t_1 * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    t_1 = 2.0d0 * (f * t_0)
    if (c <= (-9.5d-243)) then
        tmp = -(sqrt(((-16.0d0) * ((f * c) * (a * a)))) / t_0)
    else if (c <= 2.5d-66) then
        tmp = -sqrt((t_1 * (a + (c + a)))) / t_0
    else
        tmp = -(sqrt((t_1 * ((c + a) + (c + ((((b * b) * 0.5d0) / c) - a))))) / t_0)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -9.5e-243) {
		tmp = -(Math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 2.5e-66) {
		tmp = -Math.sqrt((t_1 * (A + (C + A)))) / t_0;
	} else {
		tmp = -(Math.sqrt((t_1 * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= -9.5e-243:
		tmp = -(math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0)
	elif C <= 2.5e-66:
		tmp = -math.sqrt((t_1 * (A + (C + A)))) / t_0
	else:
		tmp = -(math.sqrt((t_1 * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= -9.5e-243)
		tmp = Float64(-Float64(sqrt(Float64(-16.0 * Float64(Float64(F * C) * Float64(A * A)))) / t_0));
	elseif (C <= 2.5e-66)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(C + A))))) / t_0);
	else
		tmp = Float64(-Float64(sqrt(Float64(t_1 * Float64(Float64(C + A) + Float64(C + Float64(Float64(Float64(Float64(B * B) * 0.5) / C) - A))))) / t_0));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= -9.5e-243)
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	elseif (C <= 2.5e-66)
		tmp = -sqrt((t_1 * (A + (C + A)))) / t_0;
	else
		tmp = -(sqrt((t_1 * ((C + A) + (C + ((((B * B) * 0.5) / C) - A))))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -9.5e-243], (-N[(N[Sqrt[N[(-16.0 * N[(N[(F * C), $MachinePrecision] * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[C, 2.5e-66], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], (-N[(N[Sqrt[N[(t$95$1 * N[(N[(C + A), $MachinePrecision] + N[(C + N[(N[(N[(N[(B * B), $MachinePrecision] * 0.5), $MachinePrecision] / C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if C < -9.5000000000000005e-243

   1. Initial program 10.6%

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

      1. associate-*l* 10.6%

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

      2. unpow2 10.6%

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

      3. +-commutative 10.6%

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

      4. unpow2 10.6%

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

      5. associate-*l* 10.6%

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

      6. unpow2 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in A around inf 6.7%

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

   5. Taylor expanded in A around inf 18.6%

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

      1. unpow2 18.6%

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

   7. Simplified 18.6%

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

   if -9.5000000000000005e-243 < C < 2.49999999999999981e-66

   1. Initial program 22.3%

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

      1. associate-*l* 22.3%

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

      2. unpow2 22.3%

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

      3. +-commutative 22.3%

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

      4. unpow2 22.3%

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

      5. associate-*l* 22.3%

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

      6. unpow2 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in A around inf 15.5%

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

   if 2.49999999999999981e-66 < C

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. unpow2 28.9%

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

      3. +-commutative 28.9%

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

      4. unpow2 28.9%

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

      5. associate-*l* 28.9%

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

      6. unpow2 28.9%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in C around inf 32.0%

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

      1. mul-1-neg 32.0%

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

      2. unsub-neg 32.0%

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

   6. Simplified 32.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5 + 0}{C} - A\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.5 \cdot 10^{-243}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(C + \left(\frac{\left(B \cdot B\right) \cdot 0.5}{C} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 13: 16.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{-243}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= C -1.2e-243)
     (- (/ (sqrt (* -16.0 (* (* F C) (* A A)))) t_0))
     (if (<= C 3e-66)
       (/ (- (sqrt (* t_1 (+ A (+ C A))))) t_0)
       (- (/ (sqrt (* t_1 (+ (+ C A) (- C A)))) t_0))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -1.2e-243) {
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 3e-66) {
		tmp = -sqrt((t_1 * (A + (C + A)))) / t_0;
	} else {
		tmp = -(sqrt((t_1 * ((C + A) + (C - A)))) / t_0);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    t_1 = 2.0d0 * (f * t_0)
    if (c <= (-1.2d-243)) then
        tmp = -(sqrt(((-16.0d0) * ((f * c) * (a * a)))) / t_0)
    else if (c <= 3d-66) then
        tmp = -sqrt((t_1 * (a + (c + a)))) / t_0
    else
        tmp = -(sqrt((t_1 * ((c + a) + (c - a)))) / t_0)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (C <= -1.2e-243) {
		tmp = -(Math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 3e-66) {
		tmp = -Math.sqrt((t_1 * (A + (C + A)))) / t_0;
	} else {
		tmp = -(Math.sqrt((t_1 * ((C + A) + (C - A)))) / t_0);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if C <= -1.2e-243:
		tmp = -(math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0)
	elif C <= 3e-66:
		tmp = -math.sqrt((t_1 * (A + (C + A)))) / t_0
	else:
		tmp = -(math.sqrt((t_1 * ((C + A) + (C - A)))) / t_0)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (C <= -1.2e-243)
		tmp = Float64(-Float64(sqrt(Float64(-16.0 * Float64(Float64(F * C) * Float64(A * A)))) / t_0));
	elseif (C <= 3e-66)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + Float64(C + A))))) / t_0);
	else
		tmp = Float64(-Float64(sqrt(Float64(t_1 * Float64(Float64(C + A) + Float64(C - A)))) / t_0));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (C <= -1.2e-243)
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	elseif (C <= 3e-66)
		tmp = -sqrt((t_1 * (A + (C + A)))) / t_0;
	else
		tmp = -(sqrt((t_1 * ((C + A) + (C - A)))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.2e-243], (-N[(N[Sqrt[N[(-16.0 * N[(N[(F * C), $MachinePrecision] * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[C, 3e-66], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], (-N[(N[Sqrt[N[(t$95$1 * N[(N[(C + A), $MachinePrecision] + N[(C - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.2e-243

   1. Initial program 10.6%

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

      1. associate-*l* 10.6%

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

      2. unpow2 10.6%

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

      3. +-commutative 10.6%

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

      4. unpow2 10.6%

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

      5. associate-*l* 10.6%

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

      6. unpow2 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in A around inf 6.7%

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

   5. Taylor expanded in A around inf 18.6%

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

      1. unpow2 18.6%

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

   7. Simplified 18.6%

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

   if -1.2e-243 < C < 3.0000000000000002e-66

   1. Initial program 22.3%

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

      1. associate-*l* 22.3%

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

      2. unpow2 22.3%

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

      3. +-commutative 22.3%

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

      4. unpow2 22.3%

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

      5. associate-*l* 22.3%

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

      6. unpow2 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in A around inf 15.5%

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

   if 3.0000000000000002e-66 < C

   1. Initial program 28.9%

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

      1. associate-*l* 28.9%

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

      2. unpow2 28.9%

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

      3. +-commutative 28.9%

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

      4. unpow2 28.9%

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

      5. associate-*l* 28.9%

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

      6. unpow2 28.9%

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

   3. Simplified 28.9%

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

   4. Taylor expanded in A around -inf 31.6%

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

      1. mul-1-neg 31.6%

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

      2. sub-neg 31.6%

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

   6. Simplified 31.6%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.2 \cdot 10^{-243}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 14: 13.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -1.1 \cdot 10^{-241}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C -1.1e-241)
     (- (/ (sqrt (* -16.0 (* (* F C) (* A A)))) t_0))
     (if (<= C 3.6e+25)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A (+ C A))))) t_0)
       (/ (* 2.0 (* B (- (sqrt (* F C))))) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -1.1e-241) {
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 3.6e+25) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (C + A)))) / t_0;
	} else {
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (c <= (-1.1d-241)) then
        tmp = -(sqrt(((-16.0d0) * ((f * c) * (a * a)))) / t_0)
    else if (c <= 3.6d+25) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (a + (c + a)))) / t_0
    else
        tmp = (2.0d0 * (b * -sqrt((f * c)))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= -1.1e-241) {
		tmp = -(Math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else if (C <= 3.6e+25) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + (C + A)))) / t_0;
	} else {
		tmp = (2.0 * (B * -Math.sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= -1.1e-241:
		tmp = -(math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0)
	elif C <= 3.6e+25:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + (C + A)))) / t_0
	else:
		tmp = (2.0 * (B * -math.sqrt((F * C)))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= -1.1e-241)
		tmp = Float64(-Float64(sqrt(Float64(-16.0 * Float64(Float64(F * C) * Float64(A * A)))) / t_0));
	elseif (C <= 3.6e+25)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + Float64(C + A))))) / t_0);
	else
		tmp = Float64(Float64(2.0 * Float64(B * Float64(-sqrt(Float64(F * C))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= -1.1e-241)
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	elseif (C <= 3.6e+25)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + (C + A)))) / t_0;
	else
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.1e-241], (-N[(N[Sqrt[N[(-16.0 * N[(N[(F * C), $MachinePrecision] * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[C, 3.6e+25], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(2.0 * N[(B * (-N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.1e-241

   1. Initial program 10.6%

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

      1. associate-*l* 10.6%

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

      2. unpow2 10.6%

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

      3. +-commutative 10.6%

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

      4. unpow2 10.6%

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

      5. associate-*l* 10.6%

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

      6. unpow2 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in A around inf 6.7%

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

   5. Taylor expanded in A around inf 18.6%

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

      1. unpow2 18.6%

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

   7. Simplified 18.6%

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

   if -1.1e-241 < C < 3.60000000000000015e25

   1. Initial program 28.8%

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

      1. associate-*l* 28.8%

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

      2. unpow2 28.8%

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

      3. +-commutative 28.8%

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

      4. unpow2 28.8%

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

      5. associate-*l* 28.8%

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

      6. unpow2 28.8%

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

   3. Simplified 28.8%

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

   4. Taylor expanded in A around inf 16.0%

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

   if 3.60000000000000015e25 < C

   1. Initial program 22.4%

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

      1. associate-*l* 22.4%

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

      2. unpow2 22.4%

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

      3. +-commutative 22.4%

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

      4. unpow2 22.4%

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

      5. associate-*l* 22.4%

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

      6. unpow2 22.4%

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

   3. Simplified 22.4%

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

      1. sqrt-prod 23.9%

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

      2. *-commutative 23.9%

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

      3. *-commutative 23.9%

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

      4. associate-+l+ 23.9%

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

      5. unpow2 23.9%

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

      6. hypot-udef 39.0%

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

      7. associate-+r+ 39.0%

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

      8. +-commutative 39.0%

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

      9. associate-+r+ 39.0%

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

   5. Applied egg-rr 39.0%

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

   6. Taylor expanded in B around inf 16.0%

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

      1. associate-*l* 16.0%

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

   8. Simplified 16.0%

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

   9. Taylor expanded in A around -inf 10.1%

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

       1. associate-*l* 10.1%

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

       2. unpow2 10.1%

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

       3. rem-square-sqrt 10.2%

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

   11. Simplified 10.2%

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

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.1 \cdot 10^{-241}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;C \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 15: 10.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq 3.65 \cdot 10^{+25}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C 3.65e+25)
     (- (/ (sqrt (* -16.0 (* (* F C) (* A A)))) t_0))
     (/ (* 2.0 (* B (- (sqrt (* F C))))) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= 3.65e+25) {
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else {
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (c <= 3.65d+25) then
        tmp = -(sqrt(((-16.0d0) * ((f * c) * (a * a)))) / t_0)
    else
        tmp = (2.0d0 * (b * -sqrt((f * c)))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= 3.65e+25) {
		tmp = -(Math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	} else {
		tmp = (2.0 * (B * -Math.sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= 3.65e+25:
		tmp = -(math.sqrt((-16.0 * ((F * C) * (A * A)))) / t_0)
	else:
		tmp = (2.0 * (B * -math.sqrt((F * C)))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= 3.65e+25)
		tmp = Float64(-Float64(sqrt(Float64(-16.0 * Float64(Float64(F * C) * Float64(A * A)))) / t_0));
	else
		tmp = Float64(Float64(2.0 * Float64(B * Float64(-sqrt(Float64(F * C))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= 3.65e+25)
		tmp = -(sqrt((-16.0 * ((F * C) * (A * A)))) / t_0);
	else
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 3.65e+25], (-N[(N[Sqrt[N[(-16.0 * N[(N[(F * C), $MachinePrecision] * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(2.0 * N[(B * (-N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 3.6499999999999998e25

   1. Initial program 18.5%

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

      1. associate-*l* 18.5%

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

      2. unpow2 18.5%

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

      3. +-commutative 18.5%

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

      4. unpow2 18.5%

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

      5. associate-*l* 18.5%

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

      6. unpow2 18.5%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in A around inf 10.7%

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

   5. Taylor expanded in A around inf 13.9%

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

      1. unpow2 13.9%

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

   7. Simplified 13.9%

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

   if 3.6499999999999998e25 < C

   1. Initial program 22.4%

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

      1. associate-*l* 22.4%

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

      2. unpow2 22.4%

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

      3. +-commutative 22.4%

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

      4. unpow2 22.4%

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

      5. associate-*l* 22.4%

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

      6. unpow2 22.4%

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

   3. Simplified 22.4%

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

      1. sqrt-prod 23.9%

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

      2. *-commutative 23.9%

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

      3. *-commutative 23.9%

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

      4. associate-+l+ 23.9%

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

      5. unpow2 23.9%

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

      6. hypot-udef 39.0%

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

      7. associate-+r+ 39.0%

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

      8. +-commutative 39.0%

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

      9. associate-+r+ 39.0%

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

   5. Applied egg-rr 39.0%

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

   6. Taylor expanded in B around inf 16.0%

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

      1. associate-*l* 16.0%

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

   8. Simplified 16.0%

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

   9. Taylor expanded in A around -inf 10.1%

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

       1. associate-*l* 10.1%

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

       2. unpow2 10.1%

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

       3. rem-square-sqrt 10.2%

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

   11. Simplified 10.2%

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

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.65 \cdot 10^{+25}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 16: 6.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq 1.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(B \cdot \left(-2\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* 4.0 (* C A)))))
   (if (<= C 1.3e-32)
     (/ (* (sqrt (* F A)) (* B (- 2.0))) t_0)
     (/ (* 2.0 (* B (- (sqrt (* F C))))) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= 1.3e-32) {
		tmp = (sqrt((F * A)) * (B * -2.0)) / t_0;
	} else {
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - (4.0d0 * (c * a))
    if (c <= 1.3d-32) then
        tmp = (sqrt((f * a)) * (b * -2.0d0)) / t_0
    else
        tmp = (2.0d0 * (b * -sqrt((f * c)))) / t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = (B * B) - (4.0 * (C * A));
	double tmp;
	if (C <= 1.3e-32) {
		tmp = (Math.sqrt((F * A)) * (B * -2.0)) / t_0;
	} else {
		tmp = (2.0 * (B * -Math.sqrt((F * C)))) / t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = (B * B) - (4.0 * (C * A))
	tmp = 0
	if C <= 1.3e-32:
		tmp = (math.sqrt((F * A)) * (B * -2.0)) / t_0
	else:
		tmp = (2.0 * (B * -math.sqrt((F * C)))) / t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	tmp = 0.0
	if (C <= 1.3e-32)
		tmp = Float64(Float64(sqrt(Float64(F * A)) * Float64(B * Float64(-2.0))) / t_0);
	else
		tmp = Float64(Float64(2.0 * Float64(B * Float64(-sqrt(Float64(F * C))))) / t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B * B) - (4.0 * (C * A));
	tmp = 0.0;
	if (C <= 1.3e-32)
		tmp = (sqrt((F * A)) * (B * -2.0)) / t_0;
	else
		tmp = (2.0 * (B * -sqrt((F * C)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 1.3e-32], N[(N[(N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision] * N[(B * (-2.0)), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(2.0 * N[(B * (-N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 1.2999999999999999e-32

   1. Initial program 15.5%

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

      1. associate-*l* 15.5%

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

      2. unpow2 15.5%

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

      3. +-commutative 15.5%

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

      4. unpow2 15.5%

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

      5. associate-*l* 15.5%

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

      6. unpow2 15.5%

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

   3. Simplified 15.5%

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

      1. sqrt-prod 18.5%

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

      2. *-commutative 18.5%

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

      3. *-commutative 18.5%

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

      4. associate-+l+ 18.9%

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

      5. unpow2 18.9%

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

      6. hypot-udef 24.7%

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

      7. associate-+r+ 23.7%

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

      8. +-commutative 23.7%

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

      9. associate-+r+ 24.1%

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

   5. Applied egg-rr 24.1%

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

   6. Taylor expanded in B around inf 7.3%

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

      1. associate-*l* 7.3%

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

   8. Simplified 7.3%

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

   9. Taylor expanded in B around 0 2.6%

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

       1. unpow2 2.6%

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

       2. rem-square-sqrt 2.6%

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

   11. Simplified 2.6%

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

   if 1.2999999999999999e-32 < C

   1. Initial program 28.3%

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

      1. associate-*l* 28.3%

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

      2. unpow2 28.3%

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

      3. +-commutative 28.3%

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

      4. unpow2 28.3%

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

      5. associate-*l* 28.3%

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

      6. unpow2 28.3%

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

   3. Simplified 28.3%

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

      1. sqrt-prod 31.2%

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

      2. *-commutative 31.2%

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

      3. *-commutative 31.2%

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

      4. associate-+l+ 31.2%

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

      5. unpow2 31.2%

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

      6. hypot-udef 43.8%

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

      7. associate-+r+ 43.8%

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

      8. +-commutative 43.8%

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

      9. associate-+r+ 44.3%

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

   5. Applied egg-rr 44.3%

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

   6. Taylor expanded in B around inf 12.7%

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

      1. associate-*l* 12.7%

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

   8. Simplified 12.7%

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

   9. Taylor expanded in A around -inf 8.3%

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

       1. associate-*l* 8.3%

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

       2. unpow2 8.3%

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

       3. rem-square-sqrt 8.4%

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

   11. Simplified 8.4%

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

4. Final simplification 4.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sqrt{F \cdot A} \cdot \left(B \cdot \left(-2\right)\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(B \cdot \left(-\sqrt{F \cdot C}\right)\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 17: 4.1% accurate, 5.4× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (/ (* 2.0 (* B (- (sqrt (* F C))))) (- (* B B) (* 4.0 (* C A)))))

C

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

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (2.0d0 * (b * -sqrt((f * c)))) / ((b * b) - (4.0d0 * (c * a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.3%

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

   1. associate-*l* 19.3%

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

   2. unpow2 19.3%

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

   3. +-commutative 19.3%

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

   4. unpow2 19.3%

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

   5. associate-*l* 19.3%

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

   6. unpow2 19.3%

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

3. Simplified 19.3%

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

   1. sqrt-prod 22.3%

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

   2. *-commutative 22.3%

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

   3. *-commutative 22.3%

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

   4. associate-+l+ 22.5%

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

   5. unpow2 22.5%

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

   6. hypot-udef 30.4%

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

   7. associate-+r+ 29.6%

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

   8. +-commutative 29.6%

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

   9. associate-+r+ 30.1%

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

5. Applied egg-rr 30.1%

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

6. Taylor expanded in B around inf 8.9%

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

   1. associate-*l* 8.9%

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

8. Simplified 8.9%

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

9. Taylor expanded in A around -inf 3.3%

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

    1. associate-*l* 3.3%

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

    2. unpow2 3.3%

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

    3. rem-square-sqrt 3.4%

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

11. Simplified 3.4%

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

12. Final simplification 3.4%

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

Alternative 18: 2.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ -\sqrt{F \cdot \left(\frac{C}{A \cdot A} + \frac{1}{A}\right)} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (- (sqrt (* F (+ (/ C (* A A)) (/ 1.0 A))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return -sqrt((F * ((C / (A * A)) + (1.0 / A))));
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * ((c / (a * a)) + (1.0d0 / a))))
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	return -Math.sqrt((F * ((C / (A * A)) + (1.0 / A))));
}

Python

B = abs(B)
def code(A, B, C, F):
	return -math.sqrt((F * ((C / (A * A)) + (1.0 / A))))

Julia

B = abs(B)
function code(A, B, C, F)
	return Float64(-sqrt(Float64(F * Float64(Float64(C / Float64(A * A)) + Float64(1.0 / A)))))
end

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = -sqrt((F * ((C / (A * A)) + (1.0 / A))));
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := (-N[Sqrt[N[(F * N[(N[(C / N[(A * A), $MachinePrecision]), $MachinePrecision] + N[(1.0 / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.3%

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

   1. associate-*l* 19.3%

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

   2. unpow2 19.3%

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

   3. +-commutative 19.3%

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

   4. unpow2 19.3%

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

   5. associate-*l* 19.3%

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

   6. unpow2 19.3%

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

3. Simplified 19.3%

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

4. Taylor expanded in A around inf 7.8%

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

   1. +-commutative 7.8%

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

   2. mul-1-neg 7.8%

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

   3. unsub-neg 7.8%

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

6. Simplified 8.2%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\left(A + \left(0.5 \cdot \left(\frac{B \cdot B + 0 \cdot \left(C - \left(-C\right)\right)}{A} + \frac{B \cdot B + 0 \cdot \left(C - \left(-C\right)\right)}{\frac{A \cdot A}{C}}\right) - C\right)\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

7. Taylor expanded in B around inf 2.5%

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

   1. mul-1-neg 2.5%

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

   2. unpow2 2.5%

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

9. Simplified 2.5%

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

    1. sqrt-unprod 2.5%

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

    2. metadata-eval 2.5%

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

    3. metadata-eval 2.5%

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

11. Applied egg-rr 2.5%

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

12. Final simplification 2.5%

    \[\leadsto -\sqrt{F \cdot \left(\frac{C}{A \cdot A} + \frac{1}{A}\right)} \]

Alternative 19: 1.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ 0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{2}{\frac{C}{B}}\right) \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (* 0.25 (* (sqrt (/ F A)) (/ 2.0 (/ C B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return 0.25 * (sqrt((F / A)) * (2.0 / (C / B)));
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 0.25d0 * (sqrt((f / a)) * (2.0d0 / (c / b)))
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	return 0.25 * (Math.sqrt((F / A)) * (2.0 / (C / B)));
}

Python

B = abs(B)
def code(A, B, C, F):
	return 0.25 * (math.sqrt((F / A)) * (2.0 / (C / B)))

Julia

B = abs(B)
function code(A, B, C, F)
	return Float64(0.25 * Float64(sqrt(Float64(F / A)) * Float64(2.0 / Float64(C / B))))
end

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = 0.25 * (sqrt((F / A)) * (2.0 / (C / B)));
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(0.25 * N[(N[Sqrt[N[(F / A), $MachinePrecision]], $MachinePrecision] * N[(2.0 / N[(C / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.3%

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

   1. associate-*l* 19.3%

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

   2. unpow2 19.3%

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

   3. +-commutative 19.3%

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

   4. unpow2 19.3%

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

   5. associate-*l* 19.3%

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

   6. unpow2 19.3%

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

3. Simplified 19.3%

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

   1. sqrt-prod 22.3%

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

   2. *-commutative 22.3%

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

   3. *-commutative 22.3%

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

   4. associate-+l+ 22.5%

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

   5. unpow2 22.5%

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

   6. hypot-udef 30.4%

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

   7. associate-+r+ 29.6%

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

   8. +-commutative 29.6%

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

   9. associate-+r+ 30.1%

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

5. Applied egg-rr 30.1%

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

6. Taylor expanded in B around inf 8.9%

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

   1. associate-*l* 8.9%

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

8. Simplified 8.9%

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

9. Taylor expanded in B around 0 1.6%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2} \cdot B}{C}\right)} \]
10. Step-by-step derivation

    1. associate-/l* 1.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \color{blue}{\frac{{\left(\sqrt{2}\right)}^{2}}{\frac{C}{B}}}\right) \]

    2. unpow2 1.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{\frac{C}{B}}\right) \]

    3. rem-square-sqrt 1.6%

       \[\leadsto 0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{\color{blue}{2}}{\frac{C}{B}}\right) \]

11. Simplified 1.6%

    \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{2}{\frac{C}{B}}\right)} \]

12. Final simplification 1.6%

    \[\leadsto 0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \frac{2}{\frac{C}{B}}\right) \]

Alternative 20: 1.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \left(0.25 \cdot \frac{B \cdot 2}{A}\right) \cdot \sqrt{\frac{F}{C}} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (* (* 0.25 (/ (* B 2.0) A)) (sqrt (/ F C))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	return (0.25 * ((B * 2.0) / A)) * sqrt((F / C));
}

Fortran

NOTE: B should be positive before calling this function
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (0.25d0 * ((b * 2.0d0) / a)) * sqrt((f / c))
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	return (0.25 * ((B * 2.0) / A)) * Math.sqrt((F / C));
}

Python

B = abs(B)
def code(A, B, C, F):
	return (0.25 * ((B * 2.0) / A)) * math.sqrt((F / C))

Julia

B = abs(B)
function code(A, B, C, F)
	return Float64(Float64(0.25 * Float64(Float64(B * 2.0) / A)) * sqrt(Float64(F / C)))
end

MATLAB

B = abs(B)
function tmp = code(A, B, C, F)
	tmp = (0.25 * ((B * 2.0) / A)) * sqrt((F / C));
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(N[(0.25 * N[(N[(B * 2.0), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.3%

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

   1. associate-*l* 19.3%

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

   2. unpow2 19.3%

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

   3. +-commutative 19.3%

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

   4. unpow2 19.3%

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

   5. associate-*l* 19.3%

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

   6. unpow2 19.3%

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

3. Simplified 19.3%

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

   1. sqrt-prod 22.3%

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

   2. *-commutative 22.3%

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

   3. *-commutative 22.3%

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

   4. associate-+l+ 22.5%

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

   5. unpow2 22.5%

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

   6. hypot-udef 30.4%

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

   7. associate-+r+ 29.6%

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

   8. +-commutative 29.6%

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

   9. associate-+r+ 30.1%

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

5. Applied egg-rr 30.1%

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

6. Taylor expanded in B around inf 8.9%

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

   1. associate-*l* 8.9%

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

8. Simplified 8.9%

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

9. Taylor expanded in A around -inf 2.0%

   \[\leadsto \color{blue}{0.25 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2} \cdot B}{A} \cdot \sqrt{\frac{F}{C}}\right)} \]
10. Step-by-step derivation

    1. associate-*r* 2.0%

       \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{\left(\sqrt{2}\right)}^{2} \cdot B}{A}\right) \cdot \sqrt{\frac{F}{C}}} \]

    2. unpow2 2.0%

       \[\leadsto \left(0.25 \cdot \frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot B}{A}\right) \cdot \sqrt{\frac{F}{C}} \]

    3. rem-square-sqrt 2.0%

       \[\leadsto \left(0.25 \cdot \frac{\color{blue}{2} \cdot B}{A}\right) \cdot \sqrt{\frac{F}{C}} \]

11. Simplified 2.0%

    \[\leadsto \color{blue}{\left(0.25 \cdot \frac{2 \cdot B}{A}\right) \cdot \sqrt{\frac{F}{C}}} \]

12. Final simplification 2.0%

    \[\leadsto \left(0.25 \cdot \frac{B \cdot 2}{A}\right) \cdot \sqrt{\frac{F}{C}} \]

Reproduce

herbie shell --seed 2023229 
(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))))