ABCF->ab-angle a

Percentage Accurate: 19.7% → 58.7%

Time: 45.1s

Alternatives: 24

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: 19.7% 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: 58.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-163}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\_m\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \sqrt{2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C)))))))
        (t_1 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e-192)
     t_0
     (if (<= (pow B_m 2.0) 1e-163)
       (/
        (*
         (hypot (sqrt (* A (* C -4.0))) B_m)
         (sqrt (* (+ (+ A C) (hypot B_m (- C A))) (* 2.0 F))))
        (- (* 4.0 (* A C)) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 2e-81)
         t_0
         (if (<= (pow B_m 2.0) 5e+203)
           (/
            (*
             (sqrt (+ C (+ A (hypot (- A C) B_m))))
             (sqrt (* 2.0 (* (- (pow B_m 2.0) t_1) F))))
            (- t_1 (pow B_m 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double t_1 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-192) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 1e-163) {
		tmp = (hypot(sqrt((A * (C * -4.0))), B_m) * sqrt((((A + C) + hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e-81) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 5e+203) {
		tmp = (sqrt((C + (A + hypot((A - C), B_m)))) * sqrt((2.0 * ((pow(B_m, 2.0) - t_1) * F)))) / (t_1 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 = -2.0 * ((C / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	double t_1 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-192) {
		tmp = t_0;
	} else if (Math.pow(B_m, 2.0) <= 1e-163) {
		tmp = (Math.hypot(Math.sqrt((A * (C * -4.0))), B_m) * Math.sqrt((((A + C) + Math.hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 2e-81) {
		tmp = t_0;
	} else if (Math.pow(B_m, 2.0) <= 5e+203) {
		tmp = (Math.sqrt((C + (A + Math.hypot((A - C), B_m)))) * Math.sqrt((2.0 * ((Math.pow(B_m, 2.0) - t_1) * F)))) / (t_1 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(B_m, C))) * -Math.sqrt(F));
	}
	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 = -2.0 * ((C / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	t_1 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-192:
		tmp = t_0
	elif math.pow(B_m, 2.0) <= 1e-163:
		tmp = (math.hypot(math.sqrt((A * (C * -4.0))), B_m) * math.sqrt((((A + C) + math.hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 2e-81:
		tmp = t_0
	elif math.pow(B_m, 2.0) <= 5e+203:
		tmp = (math.sqrt((C + (A + math.hypot((A - C), B_m)))) * math.sqrt((2.0 * ((math.pow(B_m, 2.0) - t_1) * F)))) / (t_1 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(B_m, C))) * -math.sqrt(F))
	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(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	t_1 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-192)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 1e-163)
		tmp = Float64(Float64(hypot(sqrt(Float64(A * Float64(C * -4.0))), B_m) * sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(C - A))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e-81)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 5e+203)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(Float64(A - C), B_m)))) * sqrt(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)))) / Float64(t_1 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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 = -2.0 * ((C / ((B_m ^ 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	t_1 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-192)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 1e-163)
		tmp = (hypot(sqrt((A * (C * -4.0))), B_m) * sqrt((((A + C) + hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 2e-81)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 5e+203)
		tmp = (sqrt((C + (A + hypot((A - C), B_m)))) * sqrt((2.0 * (((B_m ^ 2.0) - t_1) * F)))) / (t_1 - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	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[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-192], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-163], N[(N[(N[Sqrt[N[Sqrt[N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-81], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+203], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000001e-192 or 9.99999999999999923e-164 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-81

   1. Initial program 18.3%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in C around inf 21.9%

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

   6. Taylor expanded in C around inf 16.8%

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

      1. distribute-lft1-in 16.8%

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

      2. metadata-eval 16.8%

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

   8. Simplified 16.8%

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

   9. Taylor expanded in F around 0 29.1%

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

   if 5.0000000000000001e-192 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999923e-164

   1. Initial program 61.4%

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

   3. Simplified 80.7%

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

      1. pow1/2 80.7%

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

      2. associate-*l* 80.7%

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

      3. unpow-prod-down 99.5%

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

      4. pow1/2 99.5%

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

      5. fma-undefine 99.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. unpow2 99.5%

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

      8. hypot-define 99.5%

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

      9. associate-+r+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

      2. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 1.9999999999999999e-81 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999994e203

   1. Initial program 29.4%

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

      1. expm1-log1p-u 28.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 28.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 28.1%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. hypot-define 38.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 38.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   5. Step-by-step derivation

      1. pow1/2 38.7%

         \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right)\right)}^{0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{-{\color{blue}{\left(\left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)}}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow-prod-down 55.6%

         \[\leadsto \frac{-\color{blue}{{\left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}^{0.5} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. pow1/2 55.6%

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative 55.6%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(C + A\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. expm1-log1p-u 58.8%

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

      7. pow1/2 58.8%

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

      8. *-commutative 58.8%

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

      9. *-commutative 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. associate-+l+ 59.3%

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

   8. Simplified 59.3%

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

   if 4.99999999999999994e203 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.8%

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

   3. Taylor expanded in A around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. unpow2 8.0%

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

      3. unpow2 8.0%

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

      4. hypot-define 35.2%

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

   5. Simplified 35.2%

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

      1. pow1/2 35.2%

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

      2. *-commutative 35.2%

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

      3. unpow-prod-down 47.5%

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

      4. pow1/2 47.5%

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

      5. pow1/2 47.5%

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

   7. Applied egg-rr 47.5%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-192}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{-163}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-81}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \sqrt{t\_1}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B\_m}\right)\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (*
      (sqrt
       (*
        F
        (/ (+ A (+ C (hypot B_m (- A C)))) (fma -4.0 (* A C) (pow B_m 2.0)))))
      (- (sqrt 2.0)))
     (if (<= t_3 -1e-191)
       (/ (* (sqrt (+ C (+ A (hypot (- A C) B_m)))) (sqrt t_1)) t_2)
       (if (<= t_3 INFINITY)
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))
         (*
          (expm1 (log1p (/ (sqrt 2.0) B_m)))
          (* (sqrt (+ C (hypot B_m C))) (- (sqrt F)))))))))

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 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_3 <= -1e-191) {
		tmp = (sqrt((C + (A + hypot((A - C), B_m)))) * sqrt(t_1)) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	} else {
		tmp = expm1(log1p((sqrt(2.0) / B_m))) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= -1e-191)
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + hypot(Float64(A - C), B_m)))) * sqrt(t_1)) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))));
	else
		tmp = Float64(expm1(log1p(Float64(sqrt(2.0) / B_m))) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], 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[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -1e-191], N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Exp[N[Log[1 + N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.1%

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

   3. Taylor expanded in F around 0 19.8%

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

      1. mul-1-neg 19.8%

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

   5. Simplified 70.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-191

   1. Initial program 99.2%

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

      1. expm1-log1p-u 95.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 95.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 95.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. hypot-define 95.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 95.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   5. Step-by-step derivation

      1. pow1/2 95.4%

         \[\leadsto \frac{-\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right)\right)}^{0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. *-commutative 95.4%

         \[\leadsto \frac{-{\color{blue}{\left(\left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)}}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow-prod-down 95.5%

         \[\leadsto \frac{-\color{blue}{{\left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}^{0.5} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. pow1/2 95.5%

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative 95.5%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(C + A\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{0.5}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. expm1-log1p-u 99.3%

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

      7. pow1/2 99.3%

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

      8. *-commutative 99.3%

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

      9. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. associate-+l+ 99.3%

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

   8. Simplified 99.3%

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

   if -1e-191 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.7%

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

   3. Simplified 21.7%

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

   5. Taylor expanded in C around inf 20.0%

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

   6. Taylor expanded in C around inf 18.5%

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

      1. distribute-lft1-in 18.5%

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

      2. metadata-eval 18.5%

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

   8. Simplified 18.5%

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

   9. Taylor expanded in F around 0 34.0%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 1.8%

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

      1. mul-1-neg 1.8%

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

      2. unpow2 1.8%

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

      3. unpow2 1.8%

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

      4. hypot-define 23.5%

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

   5. Simplified 23.5%

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

      1. pow1/2 23.6%

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

      2. *-commutative 23.6%

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

      3. unpow-prod-down 33.1%

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

      4. pow1/2 33.1%

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

      5. pow1/2 33.1%

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

   7. Applied egg-rr 33.1%

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

      1. expm1-log1p-u 32.7%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]

      2. expm1-undefine 3.3%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)} - 1\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]

   9. Applied egg-rr 3.3%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)} - 1\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. expm1-define 32.7%

          \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]

   11. Simplified 32.7%

       \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{B}\right)\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, t\_0\right)\\ \mathbf{if}\;B\_m \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{t\_0}, B\_m\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 10^{+94}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_2\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* A (* C -4.0)))
        (t_1
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C)))))))
        (t_2 (fma B_m B_m t_0)))
   (if (<= B_m 3.9e-91)
     t_1
     (if (<= B_m 3.1e-82)
       (/
        (*
         (hypot (sqrt t_0) B_m)
         (sqrt (* (+ (+ A C) (hypot B_m (- C A))) (* 2.0 F))))
        (- (* 4.0 (* A C)) (pow B_m 2.0)))
       (if (<= B_m 6.5e-41)
         t_1
         (if (<= B_m 1e+94)
           (/
            (* (sqrt (* F (* 2.0 t_2))) (sqrt (+ A (+ C (hypot B_m (- A C))))))
            (- t_2))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = A * (C * -4.0);
	double t_1 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double t_2 = fma(B_m, B_m, t_0);
	double tmp;
	if (B_m <= 3.9e-91) {
		tmp = t_1;
	} else if (B_m <= 3.1e-82) {
		tmp = (hypot(sqrt(t_0), B_m) * sqrt((((A + C) + hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (B_m <= 6.5e-41) {
		tmp = t_1;
	} else if (B_m <= 1e+94) {
		tmp = (sqrt((F * (2.0 * t_2))) * sqrt((A + (C + hypot(B_m, (A - C)))))) / -t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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(A * Float64(C * -4.0))
	t_1 = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	t_2 = fma(B_m, B_m, t_0)
	tmp = 0.0
	if (B_m <= 3.9e-91)
		tmp = t_1;
	elseif (B_m <= 3.1e-82)
		tmp = Float64(Float64(hypot(sqrt(t_0), B_m) * sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(C - A))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif (B_m <= 6.5e-41)
		tmp = t_1;
	elseif (B_m <= 1e+94)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_2))) * sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))) / Float64(-t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, If[LessEqual[B$95$m, 3.9e-91], t$95$1, If[LessEqual[B$95$m, 3.1e-82], N[(N[(N[Sqrt[N[Sqrt[t$95$0], $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e-41], t$95$1, If[LessEqual[B$95$m, 1e+94], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.89999999999999994e-91 or 3.1e-82 < B < 6.5000000000000004e-41

   1. Initial program 18.8%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in C around inf 20.9%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 22.7%

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

   if 3.89999999999999994e-91 < B < 3.1e-82

   1. Initial program 67.8%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow-prod-down 99.2%

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

      4. pow1/2 99.2%

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

      5. fma-undefine 99.2%

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

      6. add-sqr-sqrt 99.2%

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

      7. unpow2 99.2%

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

      8. hypot-define 99.2%

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

      9. associate-+r+ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow1/2 99.2%

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

      2. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   if 6.5000000000000004e-41 < B < 1e94

   1. Initial program 32.5%

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

   3. Simplified 48.5%

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

      1. pow1/2 48.5%

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

      2. associate-*r* 48.5%

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

      3. associate-+r+ 48.5%

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

      4. hypot-undefine 32.5%

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

      5. unpow2 32.5%

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

      6. unpow2 32.5%

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

      7. +-commutative 32.5%

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

      8. unpow-prod-down 39.2%

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

      9. *-commutative 39.2%

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

      10. pow1/2 39.2%

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

   6. Applied egg-rr 65.0%

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

      1. unpow1/2 65.0%

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

      2. associate-*l* 65.1%

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

      3. hypot-undefine 39.3%

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

      4. unpow2 39.3%

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

      5. unpow2 39.3%

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

      6. +-commutative 39.3%

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

      7. unpow2 39.3%

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

      8. unpow2 39.3%

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

      9. hypot-undefine 65.1%

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

   8. Simplified 65.1%

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

   if 1e94 < B

   1. Initial program 7.5%

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

   3. Taylor expanded in A around 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. unpow2 11.5%

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

      3. unpow2 11.5%

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

      4. hypot-define 52.5%

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

   5. Simplified 52.5%

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

      1. pow1/2 52.6%

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

      2. *-commutative 52.6%

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

      3. unpow-prod-down 71.1%

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

      4. pow1/2 71.1%

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

      5. pow1/2 71.1%

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

   7. Applied egg-rr 71.1%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 10^{+94}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 8.8 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 4.5e-159)
     t_1
     (if (<= B_m 3.9e-82)
       (/
        (sqrt
         (*
          (* 2.0 (* (- (pow B_m 2.0) t_0) F))
          (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 8.8e-41)
         t_1
         (if (<= B_m 4.2e+71)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (fma -4.0 (* A C) (pow B_m 2.0)))))
            (- (sqrt 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = (4.0 * A) * C;
	double t_1 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 4.5e-159) {
		tmp = t_1;
	} else if (B_m <= 3.9e-82) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 8.8e-41) {
		tmp = t_1;
	} else if (B_m <= 4.2e+71) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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(Float64(4.0 * A) * C)
	t_1 = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 4.5e-159)
		tmp = t_1;
	elseif (B_m <= 3.9e-82)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 8.8e-41)
		tmp = t_1;
	elseif (B_m <= 4.2e+71)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.5e-159], t$95$1, If[LessEqual[B$95$m, 3.9e-82], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.8e-41], t$95$1, If[LessEqual[B$95$m, 4.2e+71], 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[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.49999999999999989e-159 or 3.89999999999999973e-82 < B < 8.7999999999999999e-41

   1. Initial program 18.8%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in C around inf 20.3%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.1%

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

   if 4.49999999999999989e-159 < B < 3.89999999999999973e-82

   1. Initial program 28.4%

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

   3. Taylor expanded in A around -inf 30.0%

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

   if 8.7999999999999999e-41 < B < 4.19999999999999978e71

   1. Initial program 34.8%

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

   3. Taylor expanded in F around 0 42.3%

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

      1. mul-1-neg 42.3%

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

   5. Simplified 59.5%

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

   if 4.19999999999999978e71 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in A around 0 14.1%

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

      1. mul-1-neg 14.1%

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

      2. unpow2 14.1%

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

      3. unpow2 14.1%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

      1. pow1/2 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow-prod-down 71.9%

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

      4. pow1/2 71.9%

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

      5. pow1/2 71.9%

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

   7. Applied egg-rr 71.9%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\_m\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{-41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B\_m \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 1.1e-93)
     t_0
     (if (<= B_m 3.6e-82)
       (/
        (*
         (hypot (sqrt (* A (* C -4.0))) B_m)
         (sqrt (* (+ (+ A C) (hypot B_m (- C A))) (* 2.0 F))))
        (- (* 4.0 (* A C)) (pow B_m 2.0)))
       (if (<= B_m 8e-41)
         t_0
         (if (<= B_m 2.5e+68)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (fma -4.0 (* A C) (pow B_m 2.0)))))
            (- (sqrt 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 1.1e-93) {
		tmp = t_0;
	} else if (B_m <= 3.6e-82) {
		tmp = (hypot(sqrt((A * (C * -4.0))), B_m) * sqrt((((A + C) + hypot(B_m, (C - A))) * (2.0 * F)))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (B_m <= 8e-41) {
		tmp = t_0;
	} else if (B_m <= 2.5e+68) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 1.1e-93)
		tmp = t_0;
	elseif (B_m <= 3.6e-82)
		tmp = Float64(Float64(hypot(sqrt(Float64(A * Float64(C * -4.0))), B_m) * sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(C - A))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif (B_m <= 8e-41)
		tmp = t_0;
	elseif (B_m <= 2.5e+68)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.1e-93], t$95$0, If[LessEqual[B$95$m, 3.6e-82], N[(N[(N[Sqrt[N[Sqrt[N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8e-41], t$95$0, If[LessEqual[B$95$m, 2.5e+68], 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[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.09999999999999998e-93 or 3.59999999999999998e-82 < B < 8.00000000000000005e-41

   1. Initial program 18.8%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in C around inf 20.9%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 22.7%

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

   if 1.09999999999999998e-93 < B < 3.59999999999999998e-82

   1. Initial program 67.8%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

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

      2. associate-*l* 100.0%

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

      3. unpow-prod-down 99.2%

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

      4. pow1/2 99.2%

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

      5. fma-undefine 99.2%

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

      6. add-sqr-sqrt 99.2%

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

      7. unpow2 99.2%

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

      8. hypot-define 99.2%

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

      9. associate-+r+ 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. unpow1/2 99.2%

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

      2. +-commutative 99.2%

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

   8. Simplified 99.2%

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

   if 8.00000000000000005e-41 < B < 2.5000000000000002e68

   1. Initial program 34.8%

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

   3. Taylor expanded in F around 0 42.3%

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

      1. mul-1-neg 42.3%

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

   5. Simplified 59.5%

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

   if 2.5000000000000002e68 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in A around 0 14.1%

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

      1. mul-1-neg 14.1%

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

      2. unpow2 14.1%

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

      3. unpow2 14.1%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

      1. pow1/2 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow-prod-down 71.9%

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

      4. pow1/2 71.9%

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

      5. pow1/2 71.9%

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

   7. Applied egg-rr 71.9%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right) \cdot \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, C - A\right)\right) \cdot \left(2 \cdot F\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ t_2 := -2 \cdot \left(\frac{C}{t\_1} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 2.65 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (+ (pow B_m 2.0) (* -4.0 (* A C))))
        (t_2
         (*
          -2.0
          (* (/ C t_1) (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 3.9e-159)
     t_2
     (if (<= B_m 1e-83)
       (/
        (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 1e-40)
         t_2
         (if (<= B_m 2.65e+70)
           (*
            (sqrt (/ (* F (+ A (+ C (hypot B_m (- A C))))) t_1))
            (- (sqrt 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double t_2 = -2.0 * ((C / t_1) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 3.9e-159) {
		tmp = t_2;
	} else if (B_m <= 1e-83) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1e-40) {
		tmp = t_2;
	} else if (B_m <= 2.65e+70) {
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / t_1)) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 = (4.0 * A) * C;
	double t_1 = Math.pow(B_m, 2.0) + (-4.0 * (A * C));
	double t_2 = -2.0 * ((C / t_1) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 3.9e-159) {
		tmp = t_2;
	} else if (B_m <= 1e-83) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 1e-40) {
		tmp = t_2;
	} else if (B_m <= 2.65e+70) {
		tmp = Math.sqrt(((F * (A + (C + Math.hypot(B_m, (A - C))))) / t_1)) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(B_m, C))) * -Math.sqrt(F));
	}
	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 = (4.0 * A) * C
	t_1 = math.pow(B_m, 2.0) + (-4.0 * (A * C))
	t_2 = -2.0 * ((C / t_1) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	tmp = 0
	if B_m <= 3.9e-159:
		tmp = t_2
	elif B_m <= 1e-83:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 1e-40:
		tmp = t_2
	elif B_m <= 2.65e+70:
		tmp = math.sqrt(((F * (A + (C + math.hypot(B_m, (A - C))))) / t_1)) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(B_m, C))) * -math.sqrt(F))
	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(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-2.0 * Float64(Float64(C / t_1) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 3.9e-159)
		tmp = t_2;
	elseif (B_m <= 1e-83)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1e-40)
		tmp = t_2;
	elseif (B_m <= 2.65e+70)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / t_1)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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 = (4.0 * A) * C;
	t_1 = (B_m ^ 2.0) + (-4.0 * (A * C));
	t_2 = -2.0 * ((C / t_1) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	tmp = 0.0;
	if (B_m <= 3.9e-159)
		tmp = t_2;
	elseif (B_m <= 1e-83)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 1e-40)
		tmp = t_2;
	elseif (B_m <= 2.65e+70)
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / t_1)) * -sqrt(2.0);
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[(C / t$95$1), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.9e-159], t$95$2, If[LessEqual[B$95$m, 1e-83], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1e-40], t$95$2, If[LessEqual[B$95$m, 2.65e+70], N[(N[Sqrt[N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.89999999999999977e-159 or 1e-83 < B < 9.9999999999999993e-41

   1. Initial program 19.3%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in C around inf 20.2%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.0%

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

   if 3.89999999999999977e-159 < B < 1e-83

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 30.9%

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

   if 9.9999999999999993e-41 < B < 2.65e70

   1. Initial program 34.8%

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

      1. expm1-log1p-u 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. hypot-define 47.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 47.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in F around 0 42.3%

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

      1. mul-1-neg 42.3%

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

      2. unpow2 42.3%

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

      3. unpow2 42.3%

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

      4. hypot-undefine 58.8%

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

      5. cancel-sign-sub-inv 58.8%

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

      6. metadata-eval 58.8%

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

   7. Simplified 58.8%

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

   if 2.65e70 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in A around 0 14.1%

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

      1. mul-1-neg 14.1%

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

      2. unpow2 14.1%

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

      3. unpow2 14.1%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

      1. pow1/2 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow-prod-down 71.9%

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

      4. pow1/2 71.9%

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

      5. pow1/2 71.9%

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

   7. Applied egg-rr 71.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 10^{-40}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ t_2 := -2 \cdot \left(\frac{C}{t\_1} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0 \cdot \left(\left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 3.85 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (+ (pow B_m 2.0) (* -4.0 (* A C))))
        (t_2
         (*
          -2.0
          (* (/ C t_1) (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 6.5e-159)
     t_2
     (if (<= B_m 3.1e-82)
       (/
        -1.0
        (/
         t_0
         (sqrt
          (* t_0 (* (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)) (* 2.0 F))))))
       (if (<= B_m 4.5e-41)
         t_2
         (if (<= B_m 3.85e+68)
           (*
            (sqrt (/ (* F (+ A (+ C (hypot B_m (- A C))))) t_1))
            (- (sqrt 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double t_2 = -2.0 * ((C / t_1) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 6.5e-159) {
		tmp = t_2;
	} else if (B_m <= 3.1e-82) {
		tmp = -1.0 / (t_0 / sqrt((t_0 * (((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)) * (2.0 * F)))));
	} else if (B_m <= 4.5e-41) {
		tmp = t_2;
	} else if (B_m <= 3.85e+68) {
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / t_1)) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-2.0 * Float64(Float64(C / t_1) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 6.5e-159)
		tmp = t_2;
	elseif (B_m <= 3.1e-82)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_0 * Float64(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)) * Float64(2.0 * F))))));
	elseif (B_m <= 4.5e-41)
		tmp = t_2;
	elseif (B_m <= 3.85e+68)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / t_1)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[(C / t$95$1), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.5e-159], t$95$2, If[LessEqual[B$95$m, 3.1e-82], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$0 * N[(N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.5e-41], t$95$2, If[LessEqual[B$95$m, 3.85e+68], N[(N[Sqrt[N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 6.5000000000000001e-159 or 3.1e-82 < B < 4.5e-41

   1. Initial program 18.8%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in C around inf 20.3%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.1%

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

   if 6.5000000000000001e-159 < B < 3.1e-82

   1. Initial program 28.4%

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

   3. Simplified 43.9%

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

      1. clear-num 43.9%

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

      2. inv-pow 43.9%

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

   6. Applied egg-rr 43.9%

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

      1. unpow-1 43.9%

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

      2. associate-*r* 43.9%

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

      3. hypot-undefine 29.4%

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

      4. unpow2 29.4%

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

      5. unpow2 29.4%

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

      6. +-commutative 29.4%

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

      7. unpow2 29.4%

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

      8. unpow2 29.4%

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

      9. hypot-undefine 43.9%

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

   8. Simplified 43.9%

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

   9. Taylor expanded in A around -inf 30.3%

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

   if 4.5e-41 < B < 3.8499999999999999e68

   1. Initial program 34.8%

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

      1. expm1-log1p-u 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. hypot-define 47.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 47.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in F around 0 42.3%

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

      1. mul-1-neg 42.3%

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

      2. unpow2 42.3%

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

      3. unpow2 42.3%

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

      4. hypot-undefine 58.8%

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

      5. cancel-sign-sub-inv 58.8%

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

      6. metadata-eval 58.8%

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

   7. Simplified 58.8%

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

   if 3.8499999999999999e68 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in A around 0 14.1%

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

      1. mul-1-neg 14.1%

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

      2. unpow2 14.1%

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

      3. unpow2 14.1%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

      1. pow1/2 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow-prod-down 71.9%

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

      4. pow1/2 71.9%

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

      5. pow1/2 71.9%

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

   7. Applied egg-rr 71.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.85 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ t_2 := -2 \cdot \left(\frac{C}{t\_1} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 3.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 7.8 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{t\_1}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (+ (pow B_m 2.0) (* -4.0 (* A C))))
        (t_2
         (*
          -2.0
          (* (/ C t_1) (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 8.2e-159)
     t_2
     (if (<= B_m 3.6e-82)
       (/
        (sqrt
         (*
          (* 2.0 (* (- (pow B_m 2.0) t_0) F))
          (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 7.8e-41)
         t_2
         (if (<= B_m 1.25e+67)
           (*
            (sqrt (/ (* F (+ A (+ C (hypot B_m (- A C))))) t_1))
            (- (sqrt 2.0)))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt (+ C (hypot B_m C))) (- (sqrt F))))))))))

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 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double t_2 = -2.0 * ((C / t_1) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 8.2e-159) {
		tmp = t_2;
	} else if (B_m <= 3.6e-82) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 7.8e-41) {
		tmp = t_2;
	} else if (B_m <= 1.25e+67) {
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / t_1)) * -sqrt(2.0);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 = (4.0 * A) * C;
	double t_1 = Math.pow(B_m, 2.0) + (-4.0 * (A * C));
	double t_2 = -2.0 * ((C / t_1) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 8.2e-159) {
		tmp = t_2;
	} else if (B_m <= 3.6e-82) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 7.8e-41) {
		tmp = t_2;
	} else if (B_m <= 1.25e+67) {
		tmp = Math.sqrt(((F * (A + (C + Math.hypot(B_m, (A - C))))) / t_1)) * -Math.sqrt(2.0);
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(B_m, C))) * -Math.sqrt(F));
	}
	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 = (4.0 * A) * C
	t_1 = math.pow(B_m, 2.0) + (-4.0 * (A * C))
	t_2 = -2.0 * ((C / t_1) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	tmp = 0
	if B_m <= 8.2e-159:
		tmp = t_2
	elif B_m <= 3.6e-82:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 7.8e-41:
		tmp = t_2
	elif B_m <= 1.25e+67:
		tmp = math.sqrt(((F * (A + (C + math.hypot(B_m, (A - C))))) / t_1)) * -math.sqrt(2.0)
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(B_m, C))) * -math.sqrt(F))
	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(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-2.0 * Float64(Float64(C / t_1) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 8.2e-159)
		tmp = t_2;
	elseif (B_m <= 3.6e-82)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 7.8e-41)
		tmp = t_2;
	elseif (B_m <= 1.25e+67)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / t_1)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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 = (4.0 * A) * C;
	t_1 = (B_m ^ 2.0) + (-4.0 * (A * C));
	t_2 = -2.0 * ((C / t_1) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	tmp = 0.0;
	if (B_m <= 8.2e-159)
		tmp = t_2;
	elseif (B_m <= 3.6e-82)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 7.8e-41)
		tmp = t_2;
	elseif (B_m <= 1.25e+67)
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / t_1)) * -sqrt(2.0);
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(N[(C / t$95$1), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.2e-159], t$95$2, If[LessEqual[B$95$m, 3.6e-82], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.8e-41], t$95$2, If[LessEqual[B$95$m, 1.25e+67], N[(N[Sqrt[N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.20000000000000029e-159 or 3.59999999999999998e-82 < B < 7.79999999999999982e-41

   1. Initial program 18.8%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in C around inf 20.3%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.1%

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

   if 8.20000000000000029e-159 < B < 3.59999999999999998e-82

   1. Initial program 28.4%

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

   3. Taylor expanded in A around -inf 30.0%

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

   if 7.79999999999999982e-41 < B < 1.24999999999999994e67

   1. Initial program 34.8%

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

      1. expm1-log1p-u 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 33.7%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. hypot-define 47.9%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 47.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in F around 0 42.3%

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

      1. mul-1-neg 42.3%

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

      2. unpow2 42.3%

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

      3. unpow2 42.3%

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

      4. hypot-undefine 58.8%

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

      5. cancel-sign-sub-inv 58.8%

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

      6. metadata-eval 58.8%

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

   7. Simplified 58.8%

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

   if 1.24999999999999994e67 < B

   1. Initial program 8.7%

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

   3. Taylor expanded in A around 0 14.1%

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

      1. mul-1-neg 14.1%

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

      2. unpow2 14.1%

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

      3. unpow2 14.1%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

      1. pow1/2 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow-prod-down 71.9%

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

      4. pow1/2 71.9%

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

      5. pow1/2 71.9%

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

   7. Applied egg-rr 71.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 4.8e-159)
     t_1
     (if (<= B_m 9e-83)
       (/
        (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 1e-40)
         t_1
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot B_m C))) (- (sqrt F)))))))))

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 = (4.0 * A) * C;
	double t_1 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 4.8e-159) {
		tmp = t_1;
	} else if (B_m <= 9e-83) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1e-40) {
		tmp = t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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 = (4.0 * A) * C;
	double t_1 = -2.0 * ((C / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 4.8e-159) {
		tmp = t_1;
	} else if (B_m <= 9e-83) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 1e-40) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(B_m, C))) * -Math.sqrt(F));
	}
	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 = (4.0 * A) * C
	t_1 = -2.0 * ((C / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	tmp = 0
	if B_m <= 4.8e-159:
		tmp = t_1
	elif B_m <= 9e-83:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 1e-40:
		tmp = t_1
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(B_m, C))) * -math.sqrt(F))
	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(Float64(4.0 * A) * C)
	t_1 = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 4.8e-159)
		tmp = t_1;
	elseif (B_m <= 9e-83)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1e-40)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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 = (4.0 * A) * C;
	t_1 = -2.0 * ((C / ((B_m ^ 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	tmp = 0.0;
	if (B_m <= 4.8e-159)
		tmp = t_1;
	elseif (B_m <= 9e-83)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 1e-40)
		tmp = t_1;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.8e-159], t$95$1, If[LessEqual[B$95$m, 9e-83], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1e-40], t$95$1, N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.79999999999999995e-159 or 8.99999999999999995e-83 < B < 9.9999999999999993e-41

   1. Initial program 19.3%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in C around inf 20.2%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.0%

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

   if 4.79999999999999995e-159 < B < 8.99999999999999995e-83

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 30.9%

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

   if 9.9999999999999993e-41 < B

   1. Initial program 16.2%

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

   3. Taylor expanded in A around 0 19.1%

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

      1. mul-1-neg 19.1%

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

      2. unpow2 19.1%

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

      3. unpow2 19.1%

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

      4. hypot-define 48.4%

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

   5. Simplified 48.4%

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

      1. pow1/2 48.4%

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

      2. *-commutative 48.4%

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

      3. unpow-prod-down 60.5%

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

      4. pow1/2 60.5%

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

      5. pow1/2 60.5%

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

   7. Applied egg-rr 60.5%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-159}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 10^{-40}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 1.55 \cdot 10^{+128}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B\_m, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1
         (*
          -2.0
          (*
           (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
           (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))))
   (if (<= B_m 1.3e-158)
     t_1
     (if (<= B_m 8e-84)
       (/
        (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
        (- t_0 (pow B_m 2.0)))
       (if (<= B_m 6.5e-41)
         t_1
         (if (<= B_m 1.55e+128)
           (/
            -1.0
            (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (+ C (hypot B_m C))))))
           (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 = (4.0 * A) * C;
	double t_1 = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 1.3e-158) {
		tmp = t_1;
	} else if (B_m <= 8e-84) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 6.5e-41) {
		tmp = t_1;
	} else if (B_m <= 1.55e+128) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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 = (4.0 * A) * C;
	double t_1 = -2.0 * ((C / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	double tmp;
	if (B_m <= 1.3e-158) {
		tmp = t_1;
	} else if (B_m <= 8e-84) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 6.5e-41) {
		tmp = t_1;
	} else if (B_m <= 1.55e+128) {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((1.0 / F) / (C + Math.hypot(B_m, C)))));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.sqrt((1.0 / 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 = (4.0 * A) * C
	t_1 = -2.0 * ((C / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	tmp = 0
	if B_m <= 1.3e-158:
		tmp = t_1
	elif B_m <= 8e-84:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 6.5e-41:
		tmp = t_1
	elif B_m <= 1.55e+128:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((1.0 / F) / (C + math.hypot(B_m, C)))))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.sqrt((1.0 / 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(Float64(4.0 * A) * C)
	t_1 = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))))
	tmp = 0.0
	if (B_m <= 1.3e-158)
		tmp = t_1;
	elseif (B_m <= 8e-84)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 6.5e-41)
		tmp = t_1;
	elseif (B_m <= 1.55e+128)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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 = (4.0 * A) * C;
	t_1 = -2.0 * ((C / ((B_m ^ 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	tmp = 0.0;
	if (B_m <= 1.3e-158)
		tmp = t_1;
	elseif (B_m <= 8e-84)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 6.5e-41)
		tmp = t_1;
	elseif (B_m <= 1.55e+128)
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	else
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.3e-158], t$95$1, If[LessEqual[B$95$m, 8e-84], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e-41], t$95$1, If[LessEqual[B$95$m, 1.55e+128], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.3e-158 or 8.0000000000000003e-84 < B < 6.5000000000000004e-41

   1. Initial program 19.3%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in C around inf 20.2%

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

   6. Taylor expanded in C around inf 12.1%

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

      1. distribute-lft1-in 12.1%

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

      2. metadata-eval 12.1%

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

   8. Simplified 12.1%

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

   9. Taylor expanded in F around 0 24.0%

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

   if 1.3e-158 < B < 8.0000000000000003e-84

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 30.9%

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

   if 6.5000000000000004e-41 < B < 1.55000000000000002e128

   1. Initial program 36.5%

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

   3. Simplified 49.1%

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

      1. clear-num 49.2%

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

      2. inv-pow 49.2%

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

   6. Applied egg-rr 49.8%

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

      1. unpow-1 49.8%

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

      2. associate-*r* 49.8%

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

      3. hypot-undefine 36.5%

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

      4. unpow2 36.5%

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

      5. unpow2 36.5%

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

      6. +-commutative 36.5%

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

      7. unpow2 36.5%

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

      8. unpow2 36.5%

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

      9. hypot-undefine 49.8%

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

   8. Simplified 49.8%

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

   9. Taylor expanded in A around 0 40.7%

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

       1. mul-1-neg 40.7%

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

       2. associate-/r* 40.6%

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

       3. unpow2 40.6%

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

       4. unpow2 40.6%

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

       5. hypot-undefine 45.9%

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

   11. Simplified 45.9%

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

   if 1.55000000000000002e128 < B

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 38.8%

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

      1. mul-1-neg 38.8%

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

   5. Simplified 38.8%

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

      1. pow1 38.8%

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

      2. sqrt-unprod 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. unpow1 39.1%

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

   9. Simplified 39.1%

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

       1. associate-*l/ 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. pow1/2 39.1%

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

       2. div-inv 39.1%

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

       3. unpow-prod-down 65.9%

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

       4. pow1/2 65.9%

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

       5. *-commutative 65.9%

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

       6. pow1/2 65.9%

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

   13. Applied egg-rr 65.9%

       \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-158}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+128}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.18 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 7.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{B\_m \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= B_m 1.18e-94)
   (/
    -1.0
    (/
     (fma B_m B_m (* A (* C -4.0)))
     (sqrt (* -16.0 (* A (* F (pow C 2.0)))))))
   (if (<= B_m 7.6e+129)
     (/
      (* B_m (sqrt (* 2.0 (* F (+ C (hypot B_m C))))))
      (- (* (* 4.0 A) C) (pow B_m 2.0)))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (B_m <= 1.18e-94) {
		tmp = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / sqrt((-16.0 * (A * (F * pow(C, 2.0))))));
	} else if (B_m <= 7.6e+129) {
		tmp = (B_m * sqrt((2.0 * (F * (C + hypot(B_m, C)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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)
	tmp = 0.0
	if (B_m <= 1.18e-94)
		tmp = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / sqrt(Float64(-16.0 * Float64(A * Float64(F * (C ^ 2.0)))))));
	elseif (B_m <= 7.6e+129)
		tmp = Float64(Float64(B_m * sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(B_m, C)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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_] := If[LessEqual[B$95$m, 1.18e-94], N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-16.0 * N[(A * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.6e+129], N[(N[(B$95$m * N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.18e-94

   1. Initial program 18.2%

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

   3. Simplified 24.7%

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

      1. clear-num 24.7%

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

      2. inv-pow 24.7%

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

   6. Applied egg-rr 24.2%

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

      1. unpow-1 24.2%

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

      2. associate-*r* 24.2%

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

      3. hypot-undefine 18.3%

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

      4. unpow2 18.3%

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

      5. unpow2 18.3%

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

      6. +-commutative 18.3%

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

      7. unpow2 18.3%

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

      8. unpow2 18.3%

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

      9. hypot-undefine 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in A around -inf 10.8%

      \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}} \]

   if 1.18e-94 < B < 7.60000000000000011e129

   1. Initial program 36.1%

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

   3. Taylor expanded in A around 0 33.9%

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

      1. associate-*l* 33.9%

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

      2. unpow2 33.9%

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

      3. unpow2 33.9%

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

      4. hypot-define 37.8%

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

   5. Simplified 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. distribute-lft-neg-in 37.8%

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

      3. sqrt-unprod 37.8%

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

      4. *-commutative 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. *-lft-identity 37.8%

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

   9. Simplified 37.8%

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

   if 7.60000000000000011e129 < B

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 38.8%

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

      1. mul-1-neg 38.8%

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

   5. Simplified 38.8%

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

      1. pow1 38.8%

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

      2. sqrt-unprod 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. unpow1 39.1%

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

   9. Simplified 39.1%

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

       1. associate-*l/ 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. pow1/2 39.1%

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

       2. div-inv 39.1%

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

       3. unpow-prod-down 65.9%

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

       4. pow1/2 65.9%

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

       5. *-commutative 65.9%

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

       6. pow1/2 65.9%

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

   13. Applied egg-rr 65.9%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.18 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.4 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B\_m, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= B_m 7.4e-41)
   (*
    -2.0
    (*
     (/ C (+ (pow B_m 2.0) (* -4.0 (* A C))))
     (sqrt (* F (+ (* A -4.0) (/ (pow B_m 2.0) C))))))
   (if (<= B_m 1.25e+130)
     (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (+ C (hypot B_m C))))))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (B_m <= 7.4e-41) {
		tmp = -2.0 * ((C / (pow(B_m, 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + (pow(B_m, 2.0) / C)))));
	} else if (B_m <= 1.25e+130) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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 tmp;
	if (B_m <= 7.4e-41) {
		tmp = -2.0 * ((C / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * Math.sqrt((F * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))));
	} else if (B_m <= 1.25e+130) {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((1.0 / F) / (C + Math.hypot(B_m, C)))));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.sqrt((1.0 / 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):
	tmp = 0
	if B_m <= 7.4e-41:
		tmp = -2.0 * ((C / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * math.sqrt((F * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))
	elif B_m <= 1.25e+130:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((1.0 / F) / (C + math.hypot(B_m, C)))))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.sqrt((1.0 / 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)
	tmp = 0.0
	if (B_m <= 7.4e-41)
		tmp = Float64(-2.0 * Float64(Float64(C / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * sqrt(Float64(F * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C))))));
	elseif (B_m <= 1.25e+130)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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 <= 7.4e-41)
		tmp = -2.0 * ((C / ((B_m ^ 2.0) + (-4.0 * (A * C)))) * sqrt((F * ((A * -4.0) + ((B_m ^ 2.0) / C)))));
	elseif (B_m <= 1.25e+130)
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	else
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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_] := If[LessEqual[B$95$m, 7.4e-41], N[(-2.0 * N[(N[(C / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+130], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.4000000000000004e-41

   1. Initial program 19.6%

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

   3. Simplified 26.3%

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

   5. Taylor expanded in C around inf 21.2%

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

   6. Taylor expanded in C around inf 11.9%

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

      1. distribute-lft1-in 11.9%

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

      2. metadata-eval 11.9%

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

   8. Simplified 11.9%

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

   9. Taylor expanded in F around 0 22.3%

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

   if 7.4000000000000004e-41 < B < 1.2499999999999999e130

   1. Initial program 36.5%

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

   3. Simplified 49.1%

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

      1. clear-num 49.2%

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

      2. inv-pow 49.2%

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

   6. Applied egg-rr 49.8%

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

      1. unpow-1 49.8%

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

      2. associate-*r* 49.8%

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

      3. hypot-undefine 36.5%

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

      4. unpow2 36.5%

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

      5. unpow2 36.5%

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

      6. +-commutative 36.5%

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

      7. unpow2 36.5%

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

      8. unpow2 36.5%

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

      9. hypot-undefine 49.8%

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

   8. Simplified 49.8%

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

   9. Taylor expanded in A around 0 40.7%

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

       1. mul-1-neg 40.7%

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

       2. associate-/r* 40.6%

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

       3. unpow2 40.6%

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

       4. unpow2 40.6%

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

       5. hypot-undefine 45.9%

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

   11. Simplified 45.9%

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

   if 1.2499999999999999e130 < B

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 38.8%

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

      1. mul-1-neg 38.8%

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

   5. Simplified 38.8%

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

      1. pow1 38.8%

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

      2. sqrt-unprod 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. unpow1 39.1%

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

   9. Simplified 39.1%

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

       1. associate-*l/ 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. pow1/2 39.1%

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

       2. div-inv 39.1%

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

       3. unpow-prod-down 65.9%

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

       4. pow1/2 65.9%

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

       5. *-commutative 65.9%

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

       6. pow1/2 65.9%

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

   13. Applied egg-rr 65.9%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.4 \cdot 10^{-41}:\\ \;\;\;\;-2 \cdot \left(\frac{C}{{B}^{2} + -4 \cdot \left(A \cdot C\right)} \cdot \sqrt{F \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 3.9 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B\_m, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= B_m 3e-91)
   (/
    -1.0
    (/
     (fma B_m B_m (* A (* C -4.0)))
     (sqrt (* -16.0 (* A (* F (pow C 2.0)))))))
   (if (<= B_m 3.9e+133)
     (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (+ C (hypot B_m C))))))
     (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 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 tmp;
	if (B_m <= 3e-91) {
		tmp = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / sqrt((-16.0 * (A * (F * pow(C, 2.0))))));
	} else if (B_m <= 3.9e+133) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	} else {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / 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)
	tmp = 0.0
	if (B_m <= 3e-91)
		tmp = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / sqrt(Float64(-16.0 * Float64(A * Float64(F * (C ^ 2.0)))))));
	elseif (B_m <= 3.9e+133)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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_] := If[LessEqual[B$95$m, 3e-91], N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-16.0 * N[(A * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.9e+133], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.0000000000000002e-91

   1. Initial program 18.2%

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

   3. Simplified 24.7%

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

      1. clear-num 24.7%

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

      2. inv-pow 24.7%

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

   6. Applied egg-rr 24.2%

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

      1. unpow-1 24.2%

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

      2. associate-*r* 24.2%

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

      3. hypot-undefine 18.3%

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

      4. unpow2 18.3%

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

      5. unpow2 18.3%

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

      6. +-commutative 18.3%

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

      7. unpow2 18.3%

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

      8. unpow2 18.3%

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

      9. hypot-undefine 24.2%

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

   8. Simplified 24.2%

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

   9. Taylor expanded in A around -inf 10.8%

      \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}} \]

   if 3.0000000000000002e-91 < B < 3.90000000000000014e133

   1. Initial program 36.1%

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

   3. Simplified 47.1%

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

      1. clear-num 47.2%

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

      2. inv-pow 47.2%

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

   6. Applied egg-rr 47.6%

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

      1. unpow-1 47.6%

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

      2. associate-*r* 47.6%

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

      3. hypot-undefine 36.3%

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

      4. unpow2 36.3%

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

      5. unpow2 36.3%

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

      6. +-commutative 36.3%

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

      7. unpow2 36.3%

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

      8. unpow2 36.3%

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

      9. hypot-undefine 47.6%

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

   8. Simplified 47.6%

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

   9. Taylor expanded in A around 0 34.3%

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

       1. mul-1-neg 34.3%

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

       2. associate-/r* 34.3%

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

       3. unpow2 34.3%

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

       4. unpow2 34.3%

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

       5. hypot-undefine 38.0%

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

   11. Simplified 38.0%

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

   if 3.90000000000000014e133 < B

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf 38.8%

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

      1. mul-1-neg 38.8%

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

   5. Simplified 38.8%

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

      1. pow1 38.8%

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

      2. sqrt-unprod 39.1%

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

   7. Applied egg-rr 39.1%

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

      1. unpow1 39.1%

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

   9. Simplified 39.1%

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

       1. associate-*l/ 39.1%

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

   11. Applied egg-rr 39.1%

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

       1. pow1/2 39.1%

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

       2. div-inv 39.1%

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

       3. unpow-prod-down 65.9%

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

       4. pow1/2 65.9%

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

       5. *-commutative 65.9%

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

       6. pow1/2 65.9%

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

   13. Applied egg-rr 65.9%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-91}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{-16 \cdot \left(A \cdot \left(F \cdot {C}^{2}\right)\right)}}}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B\_m, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= F 2.05e+114)
   (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (+ C (hypot B_m C))))))
   (* (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 tmp;
	if (F <= 2.05e+114) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	} else {
		tmp = 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 tmp;
	if (F <= 2.05e+114) {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((1.0 / F) / (C + Math.hypot(B_m, C)))));
	} else {
		tmp = 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):
	tmp = 0
	if F <= 2.05e+114:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((1.0 / F) / (C + math.hypot(B_m, C)))))
	else:
		tmp = 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)
	tmp = 0.0
	if (F <= 2.05e+114)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(C + hypot(B_m, C))))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / 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 <= 2.05e+114)
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (C + hypot(B_m, C)))));
	else
		tmp = 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_] := If[LessEqual[F, 2.05e+114], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.05e114

   1. Initial program 21.8%

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

   3. Simplified 30.2%

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

      1. clear-num 30.2%

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

      2. inv-pow 30.2%

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

   6. Applied egg-rr 29.8%

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

      1. unpow-1 29.8%

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

      2. associate-*r* 29.8%

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

      3. hypot-undefine 21.7%

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

      4. unpow2 21.7%

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

      5. unpow2 21.7%

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

      6. +-commutative 21.7%

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

      7. unpow2 21.7%

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

      8. unpow2 21.7%

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

      9. hypot-undefine 29.8%

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

   8. Simplified 29.8%

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

   9. Taylor expanded in A around 0 10.2%

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

       1. mul-1-neg 10.2%

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

       2. associate-/r* 10.2%

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

       3. unpow2 10.2%

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

       4. unpow2 10.2%

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

       5. hypot-undefine 23.5%

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

   11. Simplified 23.5%

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

   if 2.05e114 < F

   1. Initial program 8.8%

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

   3. Taylor expanded in B around inf 18.4%

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

      1. mul-1-neg 18.4%

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

   5. Simplified 18.4%

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

      1. pow1 18.4%

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

      2. sqrt-unprod 18.5%

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

   7. Applied egg-rr 18.5%

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

      1. unpow1 18.5%

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

   9. Simplified 18.5%

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

       1. associate-*l/ 18.5%

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

   11. Applied egg-rr 18.5%

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

       1. pow1/2 19.2%

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

       2. associate-/l* 19.2%

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

       3. unpow-prod-down 20.1%

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

       4. pow1/2 20.1%

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

   13. Applied egg-rr 20.1%

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

       1. unpow1/2 20.1%

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

   15. Simplified 20.1%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{C + \mathsf{hypot}\left(B, C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= F 1.8e+114)
   (/ (sqrt (* 2.0 (* F (+ C (hypot B_m 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 tmp;
	if (F <= 1.8e+114) {
		tmp = sqrt((2.0 * (F * (C + hypot(B_m, C))))) / -B_m;
	} else {
		tmp = 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 tmp;
	if (F <= 1.8e+114) {
		tmp = Math.sqrt((2.0 * (F * (C + Math.hypot(B_m, C))))) / -B_m;
	} else {
		tmp = 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):
	tmp = 0
	if F <= 1.8e+114:
		tmp = math.sqrt((2.0 * (F * (C + math.hypot(B_m, C))))) / -B_m
	else:
		tmp = 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)
	tmp = 0.0
	if (F <= 1.8e+114)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(B_m, C))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / 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 <= 1.8e+114)
		tmp = sqrt((2.0 * (F * (C + hypot(B_m, C))))) / -B_m;
	else
		tmp = 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_] := If[LessEqual[F, 1.8e+114], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.8e114

   1. Initial program 21.8%

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

   3. Taylor expanded in A around 0 10.1%

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

      1. mul-1-neg 10.1%

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

      2. unpow2 10.1%

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

      3. unpow2 10.1%

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

      4. hypot-define 23.3%

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

   5. Simplified 23.3%

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

      1. associate-*l/ 23.3%

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

      2. sqrt-unprod 23.4%

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

   7. Applied egg-rr 23.4%

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

   if 1.8e114 < F

   1. Initial program 8.8%

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

   3. Taylor expanded in B around inf 18.4%

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

      1. mul-1-neg 18.4%

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

   5. Simplified 18.4%

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

      1. pow1 18.4%

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

      2. sqrt-unprod 18.5%

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

   7. Applied egg-rr 18.5%

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

      1. unpow1 18.5%

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

   9. Simplified 18.5%

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

       1. associate-*l/ 18.5%

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

   11. Applied egg-rr 18.5%

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

       1. pow1/2 19.2%

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

       2. associate-/l* 19.2%

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

       3. unpow-prod-down 20.1%

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

       4. pow1/2 20.1%

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

   13. Applied egg-rr 20.1%

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

       1. unpow1/2 20.1%

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

   15. Simplified 20.1%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 6e+161)
   (* (sqrt (* 2.0 F)) (- (sqrt (/ 1.0 B_m))))
   (* -2.0 (/ (sqrt (* C 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) {
	double tmp;
	if (C <= 6e+161) {
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	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 (c <= 6d+161) then
        tmp = sqrt((2.0d0 * f)) * -sqrt((1.0d0 / b_m))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    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 (C <= 6e+161) {
		tmp = Math.sqrt((2.0 * F)) * -Math.sqrt((1.0 / B_m));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / 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 C <= 6e+161:
		tmp = math.sqrt((2.0 * F)) * -math.sqrt((1.0 / B_m))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / 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 (C <= 6e+161)
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-sqrt(Float64(1.0 / B_m))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 6e+161)
		tmp = sqrt((2.0 * F)) * -sqrt((1.0 / B_m));
	else
		tmp = -2.0 * (sqrt((C * F)) / 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[C, 6e+161], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 6.00000000000000023e161

   1. Initial program 21.4%

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

   3. Taylor expanded in B around inf 15.6%

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

      1. mul-1-neg 15.6%

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

   5. Simplified 15.6%

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

      1. pow1 15.6%

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

      2. sqrt-unprod 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow1 15.7%

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

   9. Simplified 15.7%

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

       1. associate-*l/ 15.7%

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

   11. Applied egg-rr 15.7%

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

       1. pow1/2 15.9%

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

       2. div-inv 15.9%

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

       3. unpow-prod-down 21.7%

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

       4. pow1/2 21.7%

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

       5. *-commutative 21.7%

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

       6. pow1/2 21.7%

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

   13. Applied egg-rr 21.7%

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

   if 6.00000000000000023e161 < C

   1. Initial program 1.6%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in C around inf 11.6%

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

   6. Taylor expanded in C around inf 11.6%

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

      1. distribute-lft1-in 11.6%

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

      2. metadata-eval 11.6%

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

   8. Simplified 11.6%

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

   9. Taylor expanded in C around 0 14.6%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

   11. Simplified 14.6%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.4 \cdot 10^{+160}:\\ \;\;\;\;-\sqrt{\left|2 \cdot \frac{F}{B\_m}\right|}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 5.4e+160)
   (- (sqrt (fabs (* 2.0 (/ F B_m)))))
   (* -2.0 (/ (sqrt (* C 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) {
	double tmp;
	if (C <= 5.4e+160) {
		tmp = -sqrt(fabs((2.0 * (F / B_m))));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	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 (c <= 5.4d+160) then
        tmp = -sqrt(abs((2.0d0 * (f / b_m))))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    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 (C <= 5.4e+160) {
		tmp = -Math.sqrt(Math.abs((2.0 * (F / B_m))));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / 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 C <= 5.4e+160:
		tmp = -math.sqrt(math.fabs((2.0 * (F / B_m))))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / 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 (C <= 5.4e+160)
		tmp = Float64(-sqrt(abs(Float64(2.0 * Float64(F / B_m)))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 5.4e+160)
		tmp = -sqrt(abs((2.0 * (F / B_m))));
	else
		tmp = -2.0 * (sqrt((C * F)) / 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[C, 5.4e+160], (-N[Sqrt[N[Abs[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.4e160

   1. Initial program 21.4%

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

   3. Taylor expanded in B around inf 15.6%

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

      1. mul-1-neg 15.6%

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

   5. Simplified 15.6%

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

      1. pow1 15.6%

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

      2. sqrt-unprod 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow1 15.7%

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

   9. Simplified 15.7%

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

       1. add-sqr-sqrt 15.7%

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

       2. pow1/2 15.7%

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

       3. pow1/2 15.9%

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

       4. pow-prod-down 22.1%

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

       5. pow2 22.1%

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

       6. *-commutative 22.1%

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

   11. Applied egg-rr 22.1%

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

       1. unpow1/2 22.1%

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

       2. unpow2 22.1%

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

       3. rem-sqrt-square 28.5%

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

   13. Simplified 28.5%

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

   if 5.4e160 < C

   1. Initial program 1.6%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in C around inf 11.6%

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

   6. Taylor expanded in C around inf 11.6%

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

      1. distribute-lft1-in 11.6%

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

      2. metadata-eval 11.6%

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

   8. Simplified 11.6%

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

   9. Taylor expanded in C around 0 14.6%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

   11. Simplified 14.6%

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

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.4 \cdot 10^{+160}:\\ \;\;\;\;-\sqrt{\left|2 \cdot \frac{F}{B}\right|}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 2.3 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 2.3e+161)
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))
   (* -2.0 (/ (sqrt (* C 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) {
	double tmp;
	if (C <= 2.3e+161) {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	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 (c <= 2.3d+161) then
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    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 (C <= 2.3e+161) {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / 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 C <= 2.3e+161:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / 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 (C <= 2.3e+161)
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 2.3e+161)
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	else
		tmp = -2.0 * (sqrt((C * F)) / 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[C, 2.3e+161], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 2.2999999999999999e161

   1. Initial program 21.4%

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

   3. Taylor expanded in B around inf 15.6%

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

      1. mul-1-neg 15.6%

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

   5. Simplified 15.6%

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

      1. pow1 15.6%

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

      2. sqrt-unprod 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow1 15.7%

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

   9. Simplified 15.7%

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

       1. associate-*l/ 15.7%

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

   11. Applied egg-rr 15.7%

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

       1. pow1/2 15.9%

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

       2. associate-/l* 15.9%

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

       3. unpow-prod-down 21.7%

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

       4. pow1/2 21.7%

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

   13. Applied egg-rr 21.7%

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

       1. unpow1/2 21.7%

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

   15. Simplified 21.7%

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

   if 2.2999999999999999e161 < C

   1. Initial program 1.6%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in C around inf 11.6%

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

   6. Taylor expanded in C around inf 11.6%

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

      1. distribute-lft1-in 11.6%

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

      2. metadata-eval 11.6%

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

   8. Simplified 11.6%

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

   9. Taylor expanded in C around 0 14.6%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

   11. Simplified 14.6%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 2.3 \cdot 10^{+161}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 9.5e+160)
   (/ (- (sqrt (* 2.0 F))) (sqrt B_m))
   (* -2.0 (/ (sqrt (* C 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) {
	double tmp;
	if (C <= 9.5e+160) {
		tmp = -sqrt((2.0 * F)) / sqrt(B_m);
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	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 (c <= 9.5d+160) then
        tmp = -sqrt((2.0d0 * f)) / sqrt(b_m)
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    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 (C <= 9.5e+160) {
		tmp = -Math.sqrt((2.0 * F)) / Math.sqrt(B_m);
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / 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 C <= 9.5e+160:
		tmp = -math.sqrt((2.0 * F)) / math.sqrt(B_m)
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / 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 (C <= 9.5e+160)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * F))) / sqrt(B_m));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 9.5e+160)
		tmp = -sqrt((2.0 * F)) / sqrt(B_m);
	else
		tmp = -2.0 * (sqrt((C * F)) / 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[C, 9.5e+160], N[((-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]) / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 9.5000000000000006e160

   1. Initial program 21.4%

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

   3. Taylor expanded in B around inf 15.6%

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

      1. mul-1-neg 15.6%

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

   5. Simplified 15.6%

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

      1. pow1 15.6%

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

      2. sqrt-unprod 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow1 15.7%

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

   9. Simplified 15.7%

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

       1. associate-*l/ 15.7%

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

       2. sqrt-div 21.7%

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

   11. Applied egg-rr 21.7%

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

   if 9.5000000000000006e160 < C

   1. Initial program 1.6%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in C around inf 11.6%

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

   6. Taylor expanded in C around inf 11.6%

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

      1. distribute-lft1-in 11.6%

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

      2. metadata-eval 11.6%

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

   8. Simplified 11.6%

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

   9. Taylor expanded in C around 0 14.6%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

   11. Simplified 14.6%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot F}}{\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 27.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 1.5e+161)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* -2.0 (/ (sqrt (* C 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) {
	double tmp;
	if (C <= 1.5e+161) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	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 (c <= 1.5d+161) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    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 (C <= 1.5e+161) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / 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 C <= 1.5e+161:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / 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 (C <= 1.5e+161)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / 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 (C <= 1.5e+161)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = -2.0 * (sqrt((C * F)) / 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[C, 1.5e+161], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.50000000000000006e161

   1. Initial program 21.4%

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

   3. Taylor expanded in B around inf 15.6%

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

      1. mul-1-neg 15.6%

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

   5. Simplified 15.6%

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

      1. pow1 15.6%

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

      2. sqrt-unprod 15.7%

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

   7. Applied egg-rr 15.7%

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

      1. unpow1 15.7%

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

   9. Simplified 15.7%

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

       1. associate-*l/ 15.7%

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

   11. Applied egg-rr 15.7%

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

   if 1.50000000000000006e161 < C

   1. Initial program 1.6%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in C around inf 11.6%

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

   6. Taylor expanded in C around inf 11.6%

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

      1. distribute-lft1-in 11.6%

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

      2. metadata-eval 11.6%

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

   8. Simplified 11.6%

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

   9. Taylor expanded in C around 0 14.6%

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

       1. associate-*l/ 14.6%

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

       2. *-lft-identity 14.6%

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

   11. Simplified 14.6%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.9% 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 18.5%

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

3. Taylor expanded in B around inf 13.7%

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

   1. mul-1-neg 13.7%

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

5. Simplified 13.7%

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

   1. sqrt-unprod 13.8%

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

   2. pow1/2 14.0%

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

7. Applied egg-rr 14.0%

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

8. Final simplification 14.0%

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

Alternative 22: 26.9% accurate, 6.0× speedup

Math

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

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 (* F (/ 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) {
	return -sqrt((F * (2.0 / 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((f * (2.0d0 / 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((F * (2.0 / 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((F * (2.0 / 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(F * Float64(2.0 / 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((F * (2.0 / 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[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.5%

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

3. Taylor expanded in B around inf 13.7%

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

   1. mul-1-neg 13.7%

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

5. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. sqrt-unprod 13.8%

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

7. Applied egg-rr 13.8%

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

   1. unpow1 13.8%

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

9. Simplified 13.8%

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

    1. associate-*l/ 13.8%

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

11. Applied egg-rr 13.8%

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

    1. *-un-lft-identity 13.8%

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

    2. associate-/l* 13.8%

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

13. Applied egg-rr 13.8%

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

    1. *-lft-identity 13.8%

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

15. Simplified 13.8%

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

16. Final simplification 13.8%

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

Alternative 23: 26.9% 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 18.5%

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

3. Taylor expanded in B around inf 13.7%

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

   1. mul-1-neg 13.7%

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

5. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. sqrt-unprod 13.8%

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

7. Applied egg-rr 13.8%

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

   1. unpow1 13.8%

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

9. Simplified 13.8%

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

10. Final simplification 13.8%

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

Alternative 24: 26.9% 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 18.5%

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

3. Taylor expanded in B around inf 13.7%

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

   1. mul-1-neg 13.7%

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

5. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. sqrt-unprod 13.8%

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

7. Applied egg-rr 13.8%

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

   1. unpow1 13.8%

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

9. Simplified 13.8%

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

    1. associate-*l/ 13.8%

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

11. Applied egg-rr 13.8%

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

12. Final simplification 13.8%

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

Reproduce

herbie shell --seed 2024067 
(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))))