ABCF->ab-angle a

Percentage Accurate: 18.8% → 57.7%

Time: 29.7s

Alternatives: 18

Speedup: 6.0×

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: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right)} \cdot \sqrt{C}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(C \cdot A\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))) (t_1 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 5e-295)
     (/ (* (sqrt (* 2.0 (* 2.0 (* F (- (pow B_m 2.0) t_0))))) (sqrt C)) t_1)
     (if (<= (pow B_m 2.0) 2e+45)
       (/
        (*
         (sqrt (+ A (+ C (hypot (- A C) B_m))))
         (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* C A)))))))
        t_1)
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-295) {
		tmp = (sqrt((2.0 * (2.0 * (F * (pow(B_m, 2.0) - t_0))))) * sqrt(C)) / t_1;
	} else if (pow(B_m, 2.0) <= 2e+45) {
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (C * A))))))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-295) {
		tmp = (Math.sqrt((2.0 * (2.0 * (F * (Math.pow(B_m, 2.0) - t_0))))) * Math.sqrt(C)) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 2e+45) {
		tmp = (Math.sqrt((A + (C + Math.hypot((A - C), B_m)))) * Math.sqrt((F * (2.0 * (Math.pow(B_m, 2.0) - (4.0 * (C * A))))))) / t_1;
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	t_1 = t_0 - math.pow(B_m, 2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-295:
		tmp = (math.sqrt((2.0 * (2.0 * (F * (math.pow(B_m, 2.0) - t_0))))) * math.sqrt(C)) / t_1
	elif math.pow(B_m, 2.0) <= 2e+45:
		tmp = (math.sqrt((A + (C + math.hypot((A - C), B_m)))) * math.sqrt((F * (2.0 * (math.pow(B_m, 2.0) - (4.0 * (C * A))))))) / t_1
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-295)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))))) * sqrt(C)) / t_1);
	elseif ((B_m ^ 2.0) <= 2e+45)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))) * sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(C * A))))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	t_1 = t_0 - (B_m ^ 2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-295)
		tmp = (sqrt((2.0 * (2.0 * (F * ((B_m ^ 2.0) - t_0))))) * sqrt(C)) / t_1;
	elseif ((B_m ^ 2.0) <= 2e+45)
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * sqrt((F * (2.0 * ((B_m ^ 2.0) - (4.0 * (C * A))))))) / t_1;
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-295], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+45], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000008e-295

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 26.6%

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

      1. associate-*r* 26.6%

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

      2. sqrt-prod 28.8%

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

      3. *-commutative 28.8%

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

      4. *-commutative 28.8%

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

   5. Applied egg-rr 28.8%

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

   if 5.00000000000000008e-295 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e45

   1. Initial program 41.6%

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

      1. *-commutative 41.6%

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

      2. sqrt-prod 47.3%

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

      3. associate-+l+ 47.6%

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

      4. unpow2 47.6%

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

      5. unpow2 47.6%

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

      6. hypot-define 58.1%

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

      7. associate-*r* 58.1%

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

      8. associate-*l* 58.1%

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

   4. Applied egg-rr 58.1%

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

   if 1.9999999999999999e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. *-commutative 11.3%

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

      3. distribute-rgt-neg-in 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-define 28.1%

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

   5. Simplified 28.1%

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

      1. pow1/2 28.1%

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

      2. *-commutative 28.1%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 41.8%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right)} \cdot \sqrt{C}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(C \cdot A\right)\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right)} \cdot \sqrt{C}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, \left(C \cdot A\right) \cdot -4\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= (pow B_m 2.0) 5e-295)
     (/
      (* (sqrt (* 2.0 (* 2.0 (* F (- (pow B_m 2.0) t_0))))) (sqrt C))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+45)
       (/
        (*
         (sqrt (+ A (+ C (hypot (- A C) B_m))))
         (sqrt (* 2.0 (* F (fma B_m B_m (* (* C A) -4.0))))))
        (- (fma B_m B_m (* A (* C -4.0)))))
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-295) {
		tmp = (sqrt((2.0 * (2.0 * (F * (pow(B_m, 2.0) - t_0))))) * sqrt(C)) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+45) {
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * sqrt((2.0 * (F * fma(B_m, B_m, ((C * A) * -4.0)))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-295)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))))) * sqrt(C)) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+45)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))) * sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(C * A) * -4.0)))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-295], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+45], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000008e-295

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 26.6%

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

      1. associate-*r* 26.6%

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

      2. sqrt-prod 28.8%

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

      3. *-commutative 28.8%

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

      4. *-commutative 28.8%

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

   5. Applied egg-rr 28.8%

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

   if 5.00000000000000008e-295 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e45

   1. Initial program 41.6%

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

   3. Simplified 47.4%

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

      1. associate-*r* 47.4%

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

      2. associate-+r+ 46.3%

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

      3. hypot-undefine 41.6%

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

      4. unpow2 41.6%

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

      5. unpow2 41.6%

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

      6. +-commutative 41.6%

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

      7. sqrt-prod 47.3%

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

      8. *-commutative 47.3%

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

      9. associate-*r* 47.3%

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

      10. associate-+l+ 47.6%

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

   6. Applied egg-rr 58.0%

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

   if 1.9999999999999999e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. *-commutative 11.3%

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

      3. distribute-rgt-neg-in 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-define 28.1%

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

   5. Simplified 28.1%

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

      1. pow1/2 28.1%

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

      2. *-commutative 28.1%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 41.8%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right)} \cdot \sqrt{C}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right)} \cdot \sqrt{C}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+50}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= (pow B_m 2.0) 5e-85)
     (/
      (* (sqrt (* 2.0 (* 2.0 (* F (- (pow B_m 2.0) t_0))))) (sqrt C))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 1e+50)
       (*
        (sqrt
         (*
          F
          (/
           (+ A (+ C (hypot B_m (- A C))))
           (fma -4.0 (* C A) (pow B_m 2.0)))))
        (- (sqrt 2.0)))
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-85) {
		tmp = (sqrt((2.0 * (2.0 * (F * (pow(B_m, 2.0) - t_0))))) * sqrt(C)) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+50) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (C * A), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-85)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))))) * sqrt(C)) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+50)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(C * A), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-85], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+50], N[(N[Sqrt[N[(F * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e-85

   1. Initial program 22.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 25.4%

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

      1. associate-*r* 25.4%

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

      2. sqrt-prod 29.5%

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

      3. *-commutative 29.5%

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

      4. *-commutative 29.5%

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

   5. Applied egg-rr 29.5%

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

   if 5.0000000000000002e-85 < (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e50

   1. Initial program 49.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0 43.3%

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

      1. mul-1-neg 43.3%

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

      2. *-commutative 43.3%

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

      3. distribute-rgt-neg-in 43.3%

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

      4. associate-/l* 46.2%

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

      5. cancel-sign-sub-inv 46.2%

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

      6. metadata-eval 46.2%

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

      7. +-commutative 46.2%

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

   5. Simplified 53.2%

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

   if 1.0000000000000001e50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. *-commutative 11.5%

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

      3. distribute-rgt-neg-in 11.5%

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

      4. unpow2 11.5%

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

      5. unpow2 11.5%

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

      6. hypot-define 28.9%

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

   5. Simplified 28.9%

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

      1. pow1/2 28.9%

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

      2. *-commutative 28.9%

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

      3. hypot-undefine 11.6%

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

      4. unpow2 11.6%

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

      5. unpow2 11.6%

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

      6. unpow-prod-down 13.8%

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

   7. Applied egg-rr 43.0%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right)} \cdot \sqrt{C}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+50}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_2} \cdot \sqrt{C}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* F (- (pow B_m 2.0) t_0)))))
   (if (<= (pow B_m 2.0) 2e-157)
     (/ (* (sqrt (* 2.0 t_2)) (sqrt C)) t_1)
     (if (<= (pow B_m 2.0) 2e+45)
       (/ (sqrt (* t_2 (+ C (hypot B_m C)))) t_1)
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * (F * (pow(B_m, 2.0) - t_0));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-157) {
		tmp = (sqrt((2.0 * t_2)) * sqrt(C)) / t_1;
	} else if (pow(B_m, 2.0) <= 2e+45) {
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * (F * (Math.pow(B_m, 2.0) - t_0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-157) {
		tmp = (Math.sqrt((2.0 * t_2)) * Math.sqrt(C)) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 2e+45) {
		tmp = Math.sqrt((t_2 * (C + Math.hypot(B_m, C)))) / t_1;
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * (F * (math.pow(B_m, 2.0) - t_0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-157:
		tmp = (math.sqrt((2.0 * t_2)) * math.sqrt(C)) / t_1
	elif math.pow(B_m, 2.0) <= 2e+45:
		tmp = math.sqrt((t_2 * (C + math.hypot(B_m, C)))) / t_1
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-157)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_2)) * sqrt(C)) / t_1);
	elseif ((B_m ^ 2.0) <= 2e+45)
		tmp = Float64(sqrt(Float64(t_2 * Float64(C + hypot(B_m, C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (F * ((B_m ^ 2.0) - t_0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-157)
		tmp = (sqrt((2.0 * t_2)) * sqrt(C)) / t_1;
	elseif ((B_m ^ 2.0) <= 2e+45)
		tmp = sqrt((t_2 * (C + hypot(B_m, C)))) / t_1;
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-157], N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+45], N[(N[Sqrt[N[(t$95$2 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999989e-157

   1. Initial program 21.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 26.1%

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

      1. associate-*r* 26.1%

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

      2. sqrt-prod 30.4%

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

      3. *-commutative 30.4%

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

      4. *-commutative 30.4%

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

   5. Applied egg-rr 30.4%

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

   if 1.99999999999999989e-157 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e45

   1. Initial program 48.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 45.0%

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

      1. unpow2 45.0%

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

      2. unpow2 45.0%

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

      3. hypot-define 45.2%

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

   5. Simplified 45.2%

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

   if 1.9999999999999999e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. *-commutative 11.3%

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

      3. distribute-rgt-neg-in 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-define 28.1%

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

   5. Simplified 28.1%

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

      1. pow1/2 28.1%

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

      2. *-commutative 28.1%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 41.8%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right)} \cdot \sqrt{C}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right)} \cdot \sqrt{C}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= (pow B_m 2.0) 5e+20)
     (/
      (* (sqrt (* 2.0 (* 2.0 (* F (- (pow B_m 2.0) t_0))))) (sqrt C))
      (- t_0 (pow B_m 2.0)))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (pow(B_m, 2.0) <= 5e+20) {
		tmp = (sqrt((2.0 * (2.0 * (F * (pow(B_m, 2.0) - t_0))))) * sqrt(C)) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+20) {
		tmp = (Math.sqrt((2.0 * (2.0 * (F * (Math.pow(B_m, 2.0) - t_0))))) * Math.sqrt(C)) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+20:
		tmp = (math.sqrt((2.0 * (2.0 * (F * (math.pow(B_m, 2.0) - t_0))))) * math.sqrt(C)) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+20)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))))) * sqrt(C)) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+20)
		tmp = (sqrt((2.0 * (2.0 * (F * ((B_m ^ 2.0) - t_0))))) * sqrt(C)) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+20], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e20

   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. Add Preprocessing

   3. Taylor expanded in A around -inf 26.1%

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

      1. associate-*r* 26.1%

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

      2. sqrt-prod 29.3%

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

      3. *-commutative 29.3%

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

      4. *-commutative 29.3%

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

   5. Applied egg-rr 29.3%

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

   if 5e20 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. *-commutative 11.8%

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

      3. distribute-rgt-neg-in 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. hypot-define 27.9%

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

   5. Simplified 27.9%

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

      1. pow1/2 27.9%

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

      2. *-commutative 27.9%

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

      3. hypot-undefine 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. unpow-prod-down 13.9%

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

   7. Applied egg-rr 40.9%

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

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right)} \cdot \sqrt{C}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= (pow B_m 2.0) 4e+45)
     (/
      (sqrt (* (* 2.0 (* F (- (pow B_m 2.0) t_0))) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (pow(B_m, 2.0) <= 4e+45) {
		tmp = sqrt(((2.0 * (F * (pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e+45) {
		tmp = Math.sqrt(((2.0 * (F * (Math.pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(C, B_m))) * Math.sqrt(F)) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+45:
		tmp = math.sqrt(((2.0 * (F * (math.pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt((C + math.hypot(C, B_m))) * math.sqrt(F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = sqrt(((2.0 * (F * ((B_m ^ 2.0) - t_0))) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+45], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e45

   1. Initial program 29.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 27.1%

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

   if 3.9999999999999997e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. *-commutative 11.4%

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

      3. distribute-rgt-neg-in 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. hypot-define 28.3%

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

   5. Simplified 28.3%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 42.1%

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

4. Final simplification 34.5%

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

Alternative 7: 52.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B\_m}^{2} - t\_0\right)\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A))))
   (if (<= (pow B_m 2.0) 4e+45)
     (/
      (sqrt (* (* 2.0 (* F (- (pow B_m 2.0) t_0))) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (pow(B_m, 2.0) <= 4e+45) {
		tmp = sqrt(((2.0 * (F * (pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    if ((b_m ** 2.0d0) <= 4d+45) then
        tmp = sqrt(((2.0d0 * (f * ((b_m ** 2.0d0) - t_0))) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e+45) {
		tmp = Math.sqrt(((2.0 * (F * (Math.pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+45:
		tmp = math.sqrt(((2.0 * (F * (math.pow(B_m, 2.0) - t_0))) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - t_0))) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = sqrt(((2.0 * (F * ((B_m ^ 2.0) - t_0))) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+45], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e45

   1. Initial program 29.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 27.1%

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

   if 3.9999999999999997e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. *-commutative 11.4%

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

      3. distribute-rgt-neg-in 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. hypot-define 28.3%

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

   5. Simplified 28.3%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in C around 0 37.6%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(F \cdot C\right)\right) + 4 \cdot \left({B\_m}^{2} \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e+45)
   (/
    (sqrt (* C (+ (* -16.0 (* A (* F C))) (* 4.0 (* (pow B_m 2.0) F)))))
    (- (* C (* 4.0 A)) (pow B_m 2.0)))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e+45) {
		tmp = sqrt((C * ((-16.0 * (A * (F * C))) + (4.0 * (pow(B_m, 2.0) * F))))) / ((C * (4.0 * A)) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 4d+45) then
        tmp = sqrt((c * (((-16.0d0) * (a * (f * c))) + (4.0d0 * ((b_m ** 2.0d0) * f))))) / ((c * (4.0d0 * a)) - (b_m ** 2.0d0))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e+45) {
		tmp = Math.sqrt((C * ((-16.0 * (A * (F * C))) + (4.0 * (Math.pow(B_m, 2.0) * F))))) / ((C * (4.0 * A)) - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+45:
		tmp = math.sqrt((C * ((-16.0 * (A * (F * C))) + (4.0 * (math.pow(B_m, 2.0) * F))))) / ((C * (4.0 * A)) - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = Float64(sqrt(Float64(C * Float64(Float64(-16.0 * Float64(A * Float64(F * C))) + Float64(4.0 * Float64((B_m ^ 2.0) * F))))) / Float64(Float64(C * Float64(4.0 * A)) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+45)
		tmp = sqrt((C * ((-16.0 * (A * (F * C))) + (4.0 * ((B_m ^ 2.0) * F))))) / ((C * (4.0 * A)) - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+45], N[(N[Sqrt[N[(C * N[(N[(-16.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e45

   1. Initial program 29.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 27.1%

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

   4. Taylor expanded in C around 0 25.6%

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

   if 3.9999999999999997e45 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. *-commutative 11.4%

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

      3. distribute-rgt-neg-in 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. hypot-define 28.3%

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

   5. Simplified 28.3%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-undefine 11.4%

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

      4. unpow2 11.4%

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

      5. unpow2 11.4%

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

      6. unpow-prod-down 13.6%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in C around 0 37.6%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-16 \cdot \left(A \cdot \left(F \cdot C\right)\right) + 4 \cdot \left({B}^{2} \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B\_m}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(A \cdot \left(--4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-26)
   (/
    (sqrt (* (* 2.0 (* F (- (pow B_m 2.0) (* C (* 4.0 A))))) (* 2.0 C)))
    (* C (* A (- -4.0))))
   (if (<= (pow B_m 2.0) 5e+110)
     (/ (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (- B_m))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-26) {
		tmp = sqrt(((2.0 * (F * (pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	} else if (pow(B_m, 2.0) <= 5e+110) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-26) {
		tmp = Math.sqrt(((2.0 * (F * (Math.pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	} else if (Math.pow(B_m, 2.0) <= 5e+110) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-26:
		tmp = math.sqrt(((2.0 * (F * (math.pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)))
	elif math.pow(B_m, 2.0) <= 5e+110:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) / -B_m
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-26)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(C * Float64(4.0 * A))))) * Float64(2.0 * C))) / Float64(C * Float64(A * Float64(-(-4.0)))));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-26)
		tmp = sqrt(((2.0 * (F * ((B_m ^ 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-26], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+110], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-26

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 25.0%

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

   4. Taylor expanded in B around 0 24.4%

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

      1. associate-*r* 24.4%

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

      2. metadata-eval 24.4%

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

      3. distribute-lft-neg-in 24.4%

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

      4. *-commutative 24.4%

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

      5. distribute-lft-neg-in 24.4%

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

      6. metadata-eval 24.4%

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

      7. *-commutative 24.4%

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

   6. Simplified 24.4%

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

   if 2.0000000000000001e-26 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999978e110

   1. Initial program 54.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 29.5%

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

      1. mul-1-neg 29.5%

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

      2. *-commutative 29.5%

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

      3. distribute-rgt-neg-in 29.5%

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

      4. unpow2 29.5%

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

      5. unpow2 29.5%

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

      6. hypot-define 29.7%

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

   5. Simplified 29.7%

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

   7. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{{\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 29.8%

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

      2. distribute-neg-frac2 29.8%

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

      3. unpow1/2 29.8%

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

      4. associate-*r* 29.8%

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

   9. Simplified 29.8%

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

   if 4.99999999999999978e110 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. *-commutative 30.1%

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

      3. distribute-rgt-neg-in 30.1%

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

   5. Simplified 30.1%

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

      1. pow1 30.1%

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

      2. distribute-rgt-neg-out 30.1%

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

      3. pow1/2 30.1%

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

      4. pow1/2 30.1%

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

      5. pow-prod-down 30.3%

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

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 30.3%

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

      2. unpow1/2 30.3%

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

   9. Simplified 30.3%

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

       1. associate-*r/ 30.3%

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

       2. sqrt-div 39.5%

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

   11. Applied egg-rr 39.5%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(A \cdot \left(--4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -4\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-26)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* (* A (* F C)) -4.0))))
    (- (* C (* 4.0 A)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 5e+110)
     (/ (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (- B_m))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-26) {
		tmp = sqrt(((2.0 * C) * (2.0 * ((A * (F * C)) * -4.0)))) / ((C * (4.0 * A)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+110) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-26) {
		tmp = Math.sqrt(((2.0 * C) * (2.0 * ((A * (F * C)) * -4.0)))) / ((C * (4.0 * A)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 5e+110) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-26:
		tmp = math.sqrt(((2.0 * C) * (2.0 * ((A * (F * C)) * -4.0)))) / ((C * (4.0 * A)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 5e+110:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) / -B_m
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-26)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(Float64(A * Float64(F * C)) * -4.0)))) / Float64(Float64(C * Float64(4.0 * A)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-26)
		tmp = sqrt(((2.0 * C) * (2.0 * ((A * (F * C)) * -4.0)))) / ((C * (4.0 * A)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-26], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+110], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-26

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 25.0%

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

   4. Taylor expanded in B around 0 22.9%

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

      1. *-commutative 22.9%

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

   6. Simplified 22.9%

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

   if 2.0000000000000001e-26 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999978e110

   1. Initial program 54.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 29.5%

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

      1. mul-1-neg 29.5%

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

      2. *-commutative 29.5%

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

      3. distribute-rgt-neg-in 29.5%

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

      4. unpow2 29.5%

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

      5. unpow2 29.5%

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

      6. hypot-define 29.7%

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

   5. Simplified 29.7%

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

   7. Applied egg-rr 29.8%

      \[\leadsto \color{blue}{{\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 29.8%

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

      2. distribute-neg-frac2 29.8%

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

      3. unpow1/2 29.8%

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

      4. associate-*r* 29.8%

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

   9. Simplified 29.8%

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

   if 4.99999999999999978e110 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. *-commutative 30.1%

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

      3. distribute-rgt-neg-in 30.1%

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

   5. Simplified 30.1%

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

      1. pow1 30.1%

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

      2. distribute-rgt-neg-out 30.1%

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

      3. pow1/2 30.1%

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

      4. pow1/2 30.1%

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

      5. pow-prod-down 30.3%

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

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 30.3%

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

      2. unpow1/2 30.3%

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

   9. Simplified 30.3%

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

       1. associate-*r/ 30.3%

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

       2. sqrt-div 39.5%

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

   11. Applied egg-rr 39.5%

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

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(\left(A \cdot \left(F \cdot C\right)\right) \cdot -4\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F \cdot C}{{B\_m}^{2} - 4 \cdot \left(C \cdot A\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1.5e-157)
   (* -2.0 (sqrt (/ (* F C) (- (pow B_m 2.0) (* 4.0 (* C A))))))
   (if (<= (pow B_m 2.0) 5e+110)
     (/ (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (- B_m))
     (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-157) {
		tmp = -2.0 * sqrt(((F * C) / (pow(B_m, 2.0) - (4.0 * (C * A)))));
	} else if (pow(B_m, 2.0) <= 5e+110) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1.5e-157) {
		tmp = -2.0 * Math.sqrt(((F * C) / (Math.pow(B_m, 2.0) - (4.0 * (C * A)))));
	} else if (Math.pow(B_m, 2.0) <= 5e+110) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1.5e-157:
		tmp = -2.0 * math.sqrt(((F * C) / (math.pow(B_m, 2.0) - (4.0 * (C * A)))))
	elif math.pow(B_m, 2.0) <= 5e+110:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) / -B_m
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-157)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(F * C) / Float64((B_m ^ 2.0) - Float64(4.0 * Float64(C * A))))));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1.5e-157)
		tmp = -2.0 * sqrt(((F * C) / ((B_m ^ 2.0) - (4.0 * (C * A)))));
	elseif ((B_m ^ 2.0) <= 5e+110)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-157], N[(-2.0 * N[Sqrt[N[(N[(F * C), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+110], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.5e-157

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf 26.4%

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

   4. Taylor expanded in F around 0 17.7%

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

   if 1.5e-157 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999978e110

   1. Initial program 47.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 21.1%

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

      1. mul-1-neg 21.1%

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

      2. *-commutative 21.1%

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

      3. distribute-rgt-neg-in 21.1%

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

      4. unpow2 21.1%

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

      5. unpow2 21.1%

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

      6. hypot-define 21.4%

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

   5. Simplified 21.4%

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

   7. Applied egg-rr 21.5%

      \[\leadsto \color{blue}{{\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 21.5%

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

      2. distribute-neg-frac2 21.5%

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

      3. unpow1/2 21.5%

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

      4. associate-*r* 21.5%

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

   9. Simplified 21.5%

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

   if 4.99999999999999978e110 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 30.1%

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

      1. mul-1-neg 30.1%

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

      2. *-commutative 30.1%

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

      3. distribute-rgt-neg-in 30.1%

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

   5. Simplified 30.1%

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

      1. pow1 30.1%

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

      2. distribute-rgt-neg-out 30.1%

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

      3. pow1/2 30.1%

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

      4. pow1/2 30.1%

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

      5. pow-prod-down 30.3%

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

   7. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 30.3%

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

      2. unpow1/2 30.3%

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

   9. Simplified 30.3%

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

       1. associate-*r/ 30.3%

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

       2. sqrt-div 39.5%

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

   11. Applied egg-rr 39.5%

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

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F \cdot C}{{B}^{2} - 4 \cdot \left(C \cdot A\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B\_m}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(A \cdot \left(--4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+20)
   (/
    (sqrt (* (* 2.0 (* F (- (pow B_m 2.0) (* C (* 4.0 A))))) (* 2.0 C)))
    (* C (* A (- -4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+20) {
		tmp = sqrt(((2.0 * (F * (pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 5d+20) then
        tmp = sqrt(((2.0d0 * (f * ((b_m ** 2.0d0) - (c * (4.0d0 * a))))) * (2.0d0 * c))) / (c * (a * -(-4.0d0)))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e+20) {
		tmp = Math.sqrt(((2.0 * (F * (Math.pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e+20:
		tmp = math.sqrt(((2.0 * (F * (math.pow(B_m, 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+20)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(C * Float64(4.0 * A))))) * Float64(2.0 * C))) / Float64(C * Float64(A * Float64(-(-4.0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e+20)
		tmp = sqrt(((2.0 * (F * ((B_m ^ 2.0) - (C * (4.0 * A))))) * (2.0 * C))) / (C * (A * -(-4.0)));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+20], N[(N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e20

   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. Add Preprocessing

   3. Taylor expanded in A around -inf 26.1%

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

   4. Taylor expanded in B around 0 24.0%

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

      1. associate-*r* 24.0%

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

      2. metadata-eval 24.0%

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

      3. distribute-lft-neg-in 24.0%

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

      4. *-commutative 24.0%

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

      5. distribute-lft-neg-in 24.0%

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

      6. metadata-eval 24.0%

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

      7. *-commutative 24.0%

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

   6. Simplified 24.0%

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

   if 5e20 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. *-commutative 11.8%

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

      3. distribute-rgt-neg-in 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. hypot-define 27.9%

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

   5. Simplified 27.9%

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

      1. pow1/2 27.9%

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

      2. *-commutative 27.9%

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

      3. hypot-undefine 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. unpow-prod-down 13.9%

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

   7. Applied egg-rr 40.9%

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

   8. Taylor expanded in C around 0 36.4%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(A \cdot \left(--4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.16 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.16e+24)
   (/ (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (- B_m))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.16e+24) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.16e+24) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) / -B_m;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.16e+24:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) / -B_m
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.16e+24)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.16e+24)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) / -B_m;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.16e+24], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.16000000000000005e24

   1. Initial program 26.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0 7.6%

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

      1. mul-1-neg 7.6%

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

      2. *-commutative 7.6%

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

      3. distribute-rgt-neg-in 7.6%

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

      4. unpow2 7.6%

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

      5. unpow2 7.6%

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

      6. hypot-define 21.3%

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.5%

      \[\leadsto \color{blue}{{\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 21.5%

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

      2. distribute-neg-frac2 21.5%

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

      3. unpow1/2 21.4%

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

      4. associate-*r* 21.4%

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

   9. Simplified 21.4%

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

   if 1.16000000000000005e24 < F

   1. Initial program 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf 28.6%

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

      1. mul-1-neg 28.6%

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

      2. *-commutative 28.6%

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

      3. distribute-rgt-neg-in 28.6%

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

   5. Simplified 28.6%

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

      1. pow1 28.6%

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

      2. distribute-rgt-neg-out 28.6%

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

      3. pow1/2 28.6%

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

      4. pow1/2 29.0%

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

      5. pow-prod-down 29.2%

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

   7. Applied egg-rr 29.2%

      \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 29.2%

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

      2. unpow1/2 28.8%

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

   9. Simplified 28.8%

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

       1. associate-*r/ 28.8%

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

       2. sqrt-div 29.0%

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

   11. Applied egg-rr 29.0%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.16 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.3% accurate, 3.1× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in B around inf 17.7%

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

   1. mul-1-neg 17.7%

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

   2. *-commutative 17.7%

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

   3. distribute-rgt-neg-in 17.7%

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

5. Simplified 17.7%

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

   1. pow1 17.7%

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

   2. distribute-rgt-neg-out 17.7%

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

   3. pow1/2 17.7%

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

   4. pow1/2 17.8%

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

   5. pow-prod-down 17.9%

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

7. Applied egg-rr 17.9%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 17.9%

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

   2. unpow1/2 17.8%

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

9. Simplified 17.8%

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

    1. associate-*r/ 17.8%

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

    2. sqrt-div 21.8%

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

11. Applied egg-rr 21.8%

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

12. Final simplification 21.8%

    \[\leadsto \frac{\sqrt{2 \cdot F}}{-\sqrt{B}} \]
13. Add Preprocessing

Alternative 15: 27.1% accurate, 3.1× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (fabs (/ (* 2.0 F) B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[Abs[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])

Derivation

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. Add Preprocessing

3. Taylor expanded in B around inf 17.7%

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

   1. mul-1-neg 17.7%

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

   2. *-commutative 17.7%

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

   3. distribute-rgt-neg-in 17.7%

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

5. Simplified 17.7%

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

   1. pow1 17.7%

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

   2. distribute-rgt-neg-out 17.7%

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

   3. pow1/2 17.7%

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

   4. pow1/2 17.8%

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

   5. pow-prod-down 17.9%

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

7. Applied egg-rr 17.9%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 17.9%

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

   2. unpow1/2 17.8%

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

9. Simplified 17.8%

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

    1. add-sqr-sqrt 17.8%

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

    2. pow1/2 17.8%

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

    3. pow1/2 17.9%

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

    4. pow-prod-down 15.8%

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

    5. pow2 15.8%

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

11. Applied egg-rr 15.8%

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

    1. unpow1/2 15.8%

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

    2. unpow2 15.8%

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

    3. rem-sqrt-square 30.1%

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

    4. associate-*r/ 30.1%

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

13. Simplified 30.1%

    \[\leadsto -\sqrt{\color{blue}{\left|\frac{2 \cdot F}{B}\right|}} \]
14. Add Preprocessing

Alternative 16: 27.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(2 \cdot \frac{F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow((2.0 * (F / B_m)), 0.5);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow((2.0 * (F / B_m)), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow((2.0 * (F / B_m)), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(2.0 * Float64(F / B_m)) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -((2.0 * (F / B_m)) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

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. Add Preprocessing

3. Taylor expanded in B around inf 17.7%

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

   1. mul-1-neg 17.7%

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

   2. *-commutative 17.7%

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

   3. distribute-rgt-neg-in 17.7%

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

5. Simplified 17.7%

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

   1. distribute-rgt-neg-out 17.7%

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

   2. pow1/2 17.7%

      \[\leadsto -\color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}} \]

   3. pow1/2 17.8%

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

   4. pow-prod-down 17.9%

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

7. Applied egg-rr 17.9%

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

Alternative 17: 27.0% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (* 2.0 F) B_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

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. Add Preprocessing

3. Taylor expanded in B around inf 17.7%

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

   1. mul-1-neg 17.7%

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

   2. *-commutative 17.7%

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

   3. distribute-rgt-neg-in 17.7%

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

5. Simplified 17.7%

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

   1. pow1 17.7%

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

   2. distribute-rgt-neg-out 17.7%

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

   3. pow1/2 17.7%

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

   4. pow1/2 17.8%

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

   5. pow-prod-down 17.9%

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

7. Applied egg-rr 17.9%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 17.9%

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

   2. unpow1/2 17.8%

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

9. Simplified 17.8%

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

    1. *-un-lft-identity 17.8%

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

11. Applied egg-rr 17.8%

    \[\leadsto -\color{blue}{1 \cdot \sqrt{2 \cdot \frac{F}{B}}} \]
12. Step-by-step derivation

    1. *-lft-identity 17.8%

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

    2. associate-*r/ 17.8%

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

13. Simplified 17.8%

    \[\leadsto -\color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]
14. Add Preprocessing

Alternative 18: 27.0% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ F B_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

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. Add Preprocessing

3. Taylor expanded in B around inf 17.7%

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

   1. mul-1-neg 17.7%

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

   2. *-commutative 17.7%

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

   3. distribute-rgt-neg-in 17.7%

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

5. Simplified 17.7%

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

   1. pow1 17.7%

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

   2. distribute-rgt-neg-out 17.7%

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

   3. pow1/2 17.7%

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

   4. pow1/2 17.8%

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

   5. pow-prod-down 17.9%

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

7. Applied egg-rr 17.9%

   \[\leadsto \color{blue}{{\left(-{\left(2 \cdot \frac{F}{B}\right)}^{0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 17.9%

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

   2. unpow1/2 17.8%

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

9. Simplified 17.8%

   \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024099 
(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))))