ABCF->ab-angle a

Percentage Accurate: 18.5% → 53.8%

Time: 36.2s

Alternatives: 18

Speedup: 6.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% 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: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := C + \mathsf{hypot}\left(B\_m, C\right)\\ t_2 := C \cdot \left(4 \cdot A\right)\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := {B\_m}^{2} - t\_2\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_4\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot t\_1}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_0\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{t\_1}\right) \cdot \left(B\_m \cdot \sqrt{2}\right)}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (+ C (hypot B_m C)))
        (t_2 (* C (* 4.0 A)))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4 (- (pow B_m 2.0) t_2)))
   (if (<= (pow B_m 2.0) 1e-321)
     (/ (sqrt (* (* 2.0 t_4) (* 2.0 (* C F)))) t_3)
     (if (<= (pow B_m 2.0) 4e-233)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 1e-96)
         (/ (sqrt (* (* 2.0 (* t_4 F)) t_1)) t_3)
         (if (<= (pow B_m 2.0) 4e-18)
           (/
            (*
             (sqrt (* 2.0 (* F (fma B_m B_m (* -4.0 (* A C))))))
             (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
            (- (fma B_m B_m (* A (* C -4.0)))))
           (if (<= (pow B_m 2.0) 2e+98)
             (*
              (sqrt (* F (* 2.0 t_0)))
              (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) (- t_0)))
             (if (<= (pow B_m 2.0) 2e+265)
               (/ (* (* (sqrt F) (sqrt t_1)) (* B_m (sqrt 2.0))) t_3)
               (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = C + hypot(B_m, C);
	double t_2 = C * (4.0 * A);
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = pow(B_m, 2.0) - t_2;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = sqrt(((2.0 * t_4) * (2.0 * (C * F)))) / t_3;
	} else if (pow(B_m, 2.0) <= 4e-233) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 1e-96) {
		tmp = sqrt(((2.0 * (t_4 * F)) * t_1)) / t_3;
	} else if (pow(B_m, 2.0) <= 4e-18) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 2e+98) {
		tmp = sqrt((F * (2.0 * t_0))) * (sqrt(((A + C) + hypot((A - C), B_m))) / -t_0);
	} else if (pow(B_m, 2.0) <= 2e+265) {
		tmp = ((sqrt(F) * sqrt(t_1)) * (B_m * sqrt(2.0))) / t_3;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(C + hypot(B_m, C))
	t_2 = Float64(C * Float64(4.0 * A))
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64((B_m ^ 2.0) - t_2)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_4) * Float64(2.0 * Float64(C * F)))) / t_3);
	elseif ((B_m ^ 2.0) <= 4e-233)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 1e-96)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * t_1)) / t_3);
	elseif ((B_m ^ 2.0) <= 4e-18)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif ((B_m ^ 2.0) <= 2e+98)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / Float64(-t_0)));
	elseif ((B_m ^ 2.0) <= 2e+265)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(t_1)) * Float64(B_m * sqrt(2.0))) / t_3);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], N[(N[Sqrt[N[(N[(2.0 * t$95$4), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-233], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-96], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-18], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+98], N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+265], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322

   1. Initial program 24.5%

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

   3. Taylor expanded in A around -inf 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. distribute-frac-neg 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. *-lft-identity 30.4%

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

      2. distribute-neg-frac2 30.4%

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

      3. associate-*l* 30.5%

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

      4. associate-*r/ 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in C around inf 30.5%

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

      1. *-commutative 30.5%

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

   10. Simplified 30.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999983e-233

   1. Initial program 8.9%

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

   3. Taylor expanded in F around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. *-commutative 9.6%

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

      3. distribute-rgt-neg-in 9.6%

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

      4. associate-/l* 9.6%

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

      5. cancel-sign-sub-inv 9.6%

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

      6. metadata-eval 9.6%

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

      7. +-commutative 9.6%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around -inf 18.4%

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

   if 3.99999999999999983e-233 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999991e-97

   1. Initial program 52.9%

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

   3. Taylor expanded in A around 0 44.5%

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

      1. unpow2 44.5%

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

      2. unpow2 44.5%

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

      3. hypot-define 49.5%

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

   5. Simplified 49.5%

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

   if 9.9999999999999991e-97 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000003e-18

   1. Initial program 22.2%

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

   3. Simplified 32.8%

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

      1. associate-*r* 32.8%

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

      2. associate-+r+ 32.0%

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

      3. hypot-undefine 22.2%

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

      4. unpow2 22.2%

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

      5. unpow2 22.2%

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

      6. +-commutative 22.2%

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

      7. sqrt-prod 22.1%

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

      8. *-commutative 22.1%

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

      9. associate-*r* 22.1%

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

      10. associate-+l+ 22.2%

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

   6. Applied egg-rr 43.0%

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

   7. Taylor expanded in A around -inf 35.3%

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

   if 4.0000000000000003e-18 < (pow.f64 B #s(literal 2 binary64)) < 2e98

   1. Initial program 31.5%

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

   3. Simplified 42.5%

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

      1. associate-*r* 42.5%

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

      2. associate-+r+ 41.5%

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

      3. hypot-undefine 31.5%

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

      4. unpow2 31.5%

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

      5. unpow2 31.5%

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

      6. +-commutative 31.5%

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

      7. sqrt-prod 31.4%

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

      8. *-commutative 31.4%

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

      9. associate-*r* 31.4%

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

      10. associate-+l+ 31.4%

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

   6. Applied egg-rr 65.3%

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

      1. associate-/l* 65.3%

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

      2. associate-*l* 65.3%

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

      3. *-commutative 65.3%

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

      4. associate-*l* 65.3%

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

      5. *-commutative 65.3%

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

      6. *-commutative 65.3%

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

      7. associate-+r+ 65.1%

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

      8. associate-*r* 65.1%

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

      9. *-commutative 65.1%

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

      10. associate-*l* 65.1%

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

      11. *-commutative 65.1%

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

      12. *-commutative 65.1%

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

   8. Applied egg-rr 65.1%

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

   if 2e98 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000013e265

   1. Initial program 30.5%

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

   3. Taylor expanded in A around 0 16.1%

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

      1. *-commutative 16.1%

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

      2. unpow2 16.1%

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

      3. unpow2 16.1%

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

      4. hypot-define 19.2%

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

   5. Simplified 19.2%

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

      1. *-commutative 19.2%

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

      2. sqrt-prod 30.1%

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

   7. Applied egg-rr 30.1%

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

   if 2.00000000000000013e265 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

      3. distribute-rgt-neg-in 26.4%

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

   5. Simplified 26.4%

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

      1. sqrt-div 40.0%

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

   7. Applied egg-rr 40.0%

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

      1. distribute-rgt-neg-out 40.0%

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

      2. sqrt-undiv 26.4%

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

      3. sqrt-prod 26.6%

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

      4. pow1/2 26.6%

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

      5. neg-sub0 26.6%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.6%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.6%

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

       2. associate-*r/ 26.6%

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

       3. *-commutative 26.6%

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

   11. Simplified 26.6%

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

       1. sqrt-div 40.1%

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

       2. *-commutative 40.1%

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

   13. Applied egg-rr 40.1%

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

       1. *-commutative 40.1%

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

   15. Simplified 40.1%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \left(B \cdot \sqrt{2}\right)}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A)))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0)))))
   (if (<= t_1 -2e-205)
     (*
      (*
       (sqrt
        (/ (+ (+ A C) (hypot B_m (- A C))) (fma -4.0 (* A C) (pow B_m 2.0))))
       (sqrt F))
      (- (sqrt 2.0)))
     (if (<= t_1 INFINITY)
       (/
        (*
         (sqrt (* 2.0 (* F (fma B_m B_m (* -4.0 (* A C))))))
         (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- (fma B_m B_m (* A (* C -4.0)))))
       (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (t_1 <= -2e-205) {
		tmp = (sqrt((((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * sqrt(F)) * -sqrt(2.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -2e-205)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * sqrt(F)) * Float64(-sqrt(2.0)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[(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] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-205], N[(N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.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))) < -2e-205

   1. Initial program 43.5%

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

   3. Taylor expanded in F around 0 44.3%

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

      1. mul-1-neg 44.3%

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

      2. *-commutative 44.3%

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

      3. distribute-rgt-neg-in 44.3%

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

      4. associate-/l* 52.9%

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

      5. cancel-sign-sub-inv 52.9%

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

      6. metadata-eval 52.9%

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

      7. +-commutative 52.9%

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

   5. Simplified 69.3%

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

      1. pow1/2 69.3%

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

      2. *-commutative 69.3%

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

      3. unpow-prod-down 81.7%

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

      4. pow1/2 81.7%

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

      5. associate-+r+ 80.3%

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

      6. pow1/2 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -2e-205 < (/.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 24.0%

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

   3. Simplified 31.2%

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

      1. associate-*r* 31.2%

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

      2. associate-+r+ 29.6%

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

      3. hypot-undefine 24.0%

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

      4. unpow2 24.0%

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

      5. unpow2 24.0%

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

      6. +-commutative 24.0%

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

      7. sqrt-prod 25.4%

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

      8. *-commutative 25.4%

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

      9. associate-*r* 25.4%

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

      10. associate-+l+ 26.1%

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

   6. Applied egg-rr 44.5%

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

   7. Taylor expanded in A around -inf 42.9%

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

   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 B around inf 15.7%

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

      1. mul-1-neg 15.7%

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

      2. *-commutative 15.7%

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

      3. distribute-rgt-neg-in 15.7%

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

   5. Simplified 15.7%

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

      1. sqrt-div 23.6%

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

   7. Applied egg-rr 23.6%

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

      1. distribute-rgt-neg-out 23.6%

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

      2. sqrt-undiv 15.7%

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

      3. sqrt-prod 15.8%

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

      4. pow1/2 15.9%

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

      5. neg-sub0 15.9%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 15.8%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 15.8%

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

       2. associate-*r/ 15.8%

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

       3. *-commutative 15.8%

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

   11. Simplified 15.8%

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

       1. sqrt-div 23.7%

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

       2. *-commutative 23.7%

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

   13. Applied egg-rr 23.7%

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

       1. *-commutative 23.7%

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

   15. Simplified 23.7%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}} \leq -2 \cdot 10^{-205}:\\ \;\;\;\;\left(\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := C \cdot \left(4 \cdot A\right)\\ t_3 := \frac{\sqrt{\left(2 \cdot \left({B\_m}^{2} - t\_2\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, t\_1\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + t\_1}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (* -4.0 (* A C)))
        (t_2 (* C (* 4.0 A)))
        (t_3
         (/
          (sqrt (* (* 2.0 (- (pow B_m 2.0) t_2)) (* 2.0 (* C F))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-321)
     t_3
     (if (<= (pow B_m 2.0) 1e-257)
       (* (sqrt (* F (/ -0.5 A))) t_0)
       (if (<= (pow B_m 2.0) 4e+18)
         t_3
         (if (<= (pow B_m 2.0) 2e+70)
           (/
            (* (sqrt (* 2.0 (* F (fma B_m B_m t_1)))) (sqrt (* 2.0 C)))
            (- (fma B_m B_m (* A (* C -4.0)))))
           (if (<= (pow B_m 2.0) 2e+240)
             (*
              (sqrt
               (* F (/ (+ A (+ C (hypot B_m (- A C)))) (+ (pow B_m 2.0) t_1))))
              t_0)
             (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = -4.0 * (A * C);
	double t_2 = C * (4.0 * A);
	double t_3 = sqrt(((2.0 * (pow(B_m, 2.0) - t_2)) * (2.0 * (C * F)))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 1e-257) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 4e+18) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e+70) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, t_1)))) * sqrt((2.0 * C))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 2e+240) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + t_1)))) * t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(-4.0 * Float64(A * C))
	t_2 = Float64(C * Float64(4.0 * A))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64((B_m ^ 2.0) - t_2)) * Float64(2.0 * Float64(C * F)))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e+70)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, t_1)))) * sqrt(Float64(2.0 * C))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + t_1)))) * t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-257], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+18], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+70], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+240], 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[(N[Power[B$95$m, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322 or 9.9999999999999998e-258 < (pow.f64 B #s(literal 2 binary64)) < 4e18

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. distribute-frac-neg 28.5%

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

   5. Applied egg-rr 28.5%

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

      1. *-lft-identity 28.5%

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

      2. distribute-neg-frac2 28.5%

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

      3. associate-*l* 29.4%

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

      4. associate-*r/ 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in C around inf 27.5%

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

      1. *-commutative 27.5%

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

   10. Simplified 27.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-258

   1. Initial program 11.1%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. associate-/l* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   5. Simplified 13.3%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 4e18 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000015e70

   1. Initial program 24.6%

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

   3. Simplified 35.2%

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

      1. associate-*r* 35.2%

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

      2. associate-+r+ 35.2%

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

      3. hypot-undefine 24.6%

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

      4. unpow2 24.6%

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

      5. unpow2 24.6%

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

      6. +-commutative 24.6%

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

      7. sqrt-prod 24.6%

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

      8. *-commutative 24.6%

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

      9. associate-*r* 24.6%

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

      10. associate-+l+ 24.6%

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

   6. Applied egg-rr 77.8%

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

   7. Taylor expanded in A around -inf 45.6%

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

   if 2.00000000000000015e70 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e240

   1. Initial program 36.0%

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

   3. Taylor expanded in F around 0 41.9%

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

      1. mul-1-neg 41.9%

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

      2. *-commutative 41.9%

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

      3. distribute-rgt-neg-in 41.9%

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

      4. associate-/l* 53.9%

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

      5. cancel-sign-sub-inv 53.9%

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

      6. metadata-eval 53.9%

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

      7. +-commutative 53.9%

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

   5. Simplified 69.4%

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

      1. fma-undefine 69.4%

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

   7. Applied egg-rr 69.4%

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

   if 2.00000000000000003e240 < (pow.f64 B #s(literal 2 binary64))

   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 B around inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. *-commutative 25.8%

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

      3. distribute-rgt-neg-in 25.8%

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

   5. Simplified 25.8%

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

      1. sqrt-div 38.2%

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

   7. Applied egg-rr 38.2%

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

      1. distribute-rgt-neg-out 38.2%

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

      2. sqrt-undiv 25.8%

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

      3. sqrt-prod 26.0%

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

      4. pow1/2 26.0%

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

      5. neg-sub0 26.0%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.0%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.0%

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

       2. associate-*r/ 26.0%

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

       3. *-commutative 26.0%

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

   11. Simplified 26.0%

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

       1. sqrt-div 38.4%

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

       2. *-commutative 38.4%

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

   13. Applied egg-rr 38.4%

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

       1. *-commutative 38.4%

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

   15. Simplified 38.4%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C + \mathsf{hypot}\left(B\_m, C\right)\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := {B\_m}^{2} - t\_1\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_3\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot t\_0}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{t\_0}\right) \cdot \left(B\_m \cdot \sqrt{2}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ C (hypot B_m C)))
        (t_1 (* C (* 4.0 A)))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3 (- (pow B_m 2.0) t_1)))
   (if (<= (pow B_m 2.0) 1e-321)
     (/ (sqrt (* (* 2.0 t_3) (* 2.0 (* C F)))) t_2)
     (if (<= (pow B_m 2.0) 4e-233)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 2e+72)
         (/ (sqrt (* (* 2.0 (* t_3 F)) t_0)) t_2)
         (if (<= (pow B_m 2.0) 2e+265)
           (/ (* (* (sqrt F) (sqrt t_0)) (* B_m (sqrt 2.0))) t_2)
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C + hypot(B_m, C);
	double t_1 = C * (4.0 * A);
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = pow(B_m, 2.0) - t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	} else if (pow(B_m, 2.0) <= 4e-233) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 2e+72) {
		tmp = sqrt(((2.0 * (t_3 * F)) * t_0)) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+265) {
		tmp = ((sqrt(F) * sqrt(t_0)) * (B_m * sqrt(2.0))) / t_2;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = C + Math.hypot(B_m, C);
	double t_1 = C * (4.0 * A);
	double t_2 = t_1 - Math.pow(B_m, 2.0);
	double t_3 = Math.pow(B_m, 2.0) - t_1;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = Math.sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 4e-233) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 2e+72) {
		tmp = Math.sqrt(((2.0 * (t_3 * F)) * t_0)) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+265) {
		tmp = ((Math.sqrt(F) * Math.sqrt(t_0)) * (B_m * Math.sqrt(2.0))) / t_2;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C + math.hypot(B_m, C)
	t_1 = C * (4.0 * A)
	t_2 = t_1 - math.pow(B_m, 2.0)
	t_3 = math.pow(B_m, 2.0) - t_1
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = math.sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2
	elif math.pow(B_m, 2.0) <= 4e-233:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 2e+72:
		tmp = math.sqrt(((2.0 * (t_3 * F)) * t_0)) / t_2
	elif math.pow(B_m, 2.0) <= 2e+265:
		tmp = ((math.sqrt(F) * math.sqrt(t_0)) * (B_m * math.sqrt(2.0))) / t_2
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C + hypot(B_m, C))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64((B_m ^ 2.0) - t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_3) * Float64(2.0 * Float64(C * F)))) / t_2);
	elseif ((B_m ^ 2.0) <= 4e-233)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 2e+72)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * t_0)) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+265)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(t_0)) * Float64(B_m * sqrt(2.0))) / t_2);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C + hypot(B_m, C);
	t_1 = C * (4.0 * A);
	t_2 = t_1 - (B_m ^ 2.0);
	t_3 = (B_m ^ 2.0) - t_1;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	elseif ((B_m ^ 2.0) <= 4e-233)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 2e+72)
		tmp = sqrt(((2.0 * (t_3 * F)) * t_0)) / t_2;
	elseif ((B_m ^ 2.0) <= 2e+265)
		tmp = ((sqrt(F) * sqrt(t_0)) * (B_m * sqrt(2.0))) / t_2;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], N[(N[Sqrt[N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-233], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+72], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+265], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322

   1. Initial program 24.5%

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

   3. Taylor expanded in A around -inf 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. distribute-frac-neg 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. *-lft-identity 30.4%

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

      2. distribute-neg-frac2 30.4%

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

      3. associate-*l* 30.5%

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

      4. associate-*r/ 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in C around inf 30.5%

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

      1. *-commutative 30.5%

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

   10. Simplified 30.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999983e-233

   1. Initial program 8.9%

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

   3. Taylor expanded in F around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. *-commutative 9.6%

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

      3. distribute-rgt-neg-in 9.6%

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

      4. associate-/l* 9.6%

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

      5. cancel-sign-sub-inv 9.6%

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

      6. metadata-eval 9.6%

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

      7. +-commutative 9.6%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around -inf 18.4%

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

   if 3.99999999999999983e-233 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999989e72

   1. Initial program 35.3%

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

   3. Taylor expanded in A around 0 29.5%

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

      1. unpow2 29.5%

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

      2. unpow2 29.5%

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

      3. hypot-define 36.4%

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

   5. Simplified 36.4%

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

   if 1.99999999999999989e72 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000013e265

   1. Initial program 31.6%

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

   3. Taylor expanded in A around 0 18.0%

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

      1. *-commutative 18.0%

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

      2. unpow2 18.0%

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

      3. unpow2 18.0%

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

      4. hypot-define 21.0%

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

   5. Simplified 21.0%

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

      1. *-commutative 21.0%

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

      2. sqrt-prod 31.3%

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

   7. Applied egg-rr 31.3%

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

   if 2.00000000000000013e265 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

      3. distribute-rgt-neg-in 26.4%

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

   5. Simplified 26.4%

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

      1. sqrt-div 40.0%

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

   7. Applied egg-rr 40.0%

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

      1. distribute-rgt-neg-out 40.0%

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

      2. sqrt-undiv 26.4%

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

      3. sqrt-prod 26.6%

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

      4. pow1/2 26.6%

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

      5. neg-sub0 26.6%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.6%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.6%

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

       2. associate-*r/ 26.6%

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

       3. *-commutative 26.6%

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

   11. Simplified 26.6%

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

       1. sqrt-div 40.1%

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

       2. *-commutative 40.1%

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

   13. Applied egg-rr 40.1%

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

       1. *-commutative 40.1%

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

   15. Simplified 40.1%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \left(B \cdot \sqrt{2}\right)}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_0\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \left(B\_m \cdot \sqrt{2}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (* C (* 4.0 A)))
        (t_2 (- t_1 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 1e-321)
     (/ (sqrt (* (* 2.0 (- (pow B_m 2.0) t_1)) (* 2.0 (* C F)))) t_2)
     (if (<= (pow B_m 2.0) 2e-222)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 2e+98)
         (*
          (sqrt (* F (* 2.0 t_0)))
          (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) (- t_0)))
         (if (<= (pow B_m 2.0) 2e+265)
           (/
            (* (* (sqrt F) (sqrt (+ C (hypot B_m C)))) (* B_m (sqrt 2.0)))
            t_2)
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = C * (4.0 * A);
	double t_2 = t_1 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = sqrt(((2.0 * (pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e-222) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 2e+98) {
		tmp = sqrt((F * (2.0 * t_0))) * (sqrt(((A + C) + hypot((A - C), B_m))) / -t_0);
	} else if (pow(B_m, 2.0) <= 2e+265) {
		tmp = ((sqrt(F) * sqrt((C + hypot(B_m, C)))) * (B_m * sqrt(2.0))) / t_2;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64((B_m ^ 2.0) - t_1)) * Float64(2.0 * Float64(C * F)))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e-222)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 2e+98)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / Float64(-t_0)));
	elseif ((B_m ^ 2.0) <= 2e+265)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(B_m * sqrt(2.0))) / t_2);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], N[(N[Sqrt[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-222], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+98], N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+265], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322

   1. Initial program 24.5%

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

   3. Taylor expanded in A around -inf 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. distribute-frac-neg 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. *-lft-identity 30.4%

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

      2. distribute-neg-frac2 30.4%

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

      3. associate-*l* 30.5%

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

      4. associate-*r/ 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in C around inf 30.5%

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

      1. *-commutative 30.5%

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

   10. Simplified 30.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-222

   1. Initial program 11.6%

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

   3. Taylor expanded in F around 0 12.2%

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

      1. mul-1-neg 12.2%

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

      2. *-commutative 12.2%

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

      3. distribute-rgt-neg-in 12.2%

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

      4. associate-/l* 12.2%

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

      5. cancel-sign-sub-inv 12.2%

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

      6. metadata-eval 12.2%

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

      7. +-commutative 12.2%

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

   5. Simplified 20.6%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 2.0000000000000001e-222 < (pow.f64 B #s(literal 2 binary64)) < 2e98

   1. Initial program 35.3%

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

   3. Simplified 46.4%

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

      1. associate-*r* 46.4%

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

      2. associate-+r+ 45.6%

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

      3. hypot-undefine 35.3%

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

      4. unpow2 35.3%

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

      5. unpow2 35.3%

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

      6. +-commutative 35.3%

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

      7. sqrt-prod 35.2%

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

      8. *-commutative 35.2%

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

      9. associate-*r* 35.2%

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

      10. associate-+l+ 35.2%

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

   6. Applied egg-rr 57.0%

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

      1. associate-/l* 57.0%

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

      2. associate-*l* 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*l* 57.0%

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

      5. *-commutative 57.0%

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

      6. *-commutative 57.0%

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

      7. associate-+r+ 56.4%

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

      8. associate-*r* 56.4%

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

      9. *-commutative 56.4%

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

      10. associate-*l* 56.4%

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

      11. *-commutative 56.4%

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

      12. *-commutative 56.4%

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

   8. Applied egg-rr 56.4%

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

   if 2e98 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000013e265

   1. Initial program 30.5%

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

   3. Taylor expanded in A around 0 16.1%

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

      1. *-commutative 16.1%

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

      2. unpow2 16.1%

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

      3. unpow2 16.1%

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

      4. hypot-define 19.2%

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

   5. Simplified 19.2%

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

      1. *-commutative 19.2%

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

      2. sqrt-prod 30.1%

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

   7. Applied egg-rr 30.1%

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

   if 2.00000000000000013e265 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.7%

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

   3. Taylor expanded in B around inf 26.4%

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

      1. mul-1-neg 26.4%

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

      2. *-commutative 26.4%

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

      3. distribute-rgt-neg-in 26.4%

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

   5. Simplified 26.4%

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

      1. sqrt-div 40.0%

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

   7. Applied egg-rr 40.0%

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

      1. distribute-rgt-neg-out 40.0%

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

      2. sqrt-undiv 26.4%

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

      3. sqrt-prod 26.6%

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

      4. pow1/2 26.6%

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

      5. neg-sub0 26.6%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.6%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.6%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.6%

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

       2. associate-*r/ 26.6%

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

       3. *-commutative 26.6%

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

   11. Simplified 26.6%

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

       1. sqrt-div 40.1%

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

       2. *-commutative 40.1%

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

   13. Applied egg-rr 40.1%

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

       1. *-commutative 40.1%

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

   15. Simplified 40.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \left(B \cdot \sqrt{2}\right)}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := \frac{\sqrt{\left(2 \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (* C (* 4.0 A)))
        (t_2
         (/
          (sqrt (* (* 2.0 (- (pow B_m 2.0) t_1)) (* 2.0 (* C F))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-321)
     t_2
     (if (<= (pow B_m 2.0) 1e-257)
       (* (sqrt (* F (/ -0.5 A))) t_0)
       (if (<= (pow B_m 2.0) 4e+18)
         t_2
         (if (<= (pow B_m 2.0) 2e+240)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (+ (pow B_m 2.0) (* -4.0 (* A C))))))
            t_0)
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = sqrt(((2.0 * (pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 1e-257) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 4e+18) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 2e+240) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = Math.sqrt(((2.0 * (Math.pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - Math.pow(B_m, 2.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e-257) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 4e+18) {
		tmp = t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+240) {
		tmp = Math.sqrt((F * ((A + (C + Math.hypot(B_m, (A - C)))) / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = C * (4.0 * A)
	t_2 = math.sqrt(((2.0 * (math.pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - math.pow(B_m, 2.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = t_2
	elif math.pow(B_m, 2.0) <= 1e-257:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	elif math.pow(B_m, 2.0) <= 4e+18:
		tmp = t_2
	elif math.pow(B_m, 2.0) <= 2e+240:
		tmp = math.sqrt((F * ((A + (C + math.hypot(B_m, (A - C)))) / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64((B_m ^ 2.0) - t_1)) * Float64(2.0 * Float64(C * F)))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))))) * t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	t_1 = C * (4.0 * A);
	t_2 = sqrt(((2.0 * ((B_m ^ 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - (B_m ^ 2.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / ((B_m ^ 2.0) + (-4.0 * (A * C)))))) * t_0;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-257], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+18], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+240], 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[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322 or 9.9999999999999998e-258 < (pow.f64 B #s(literal 2 binary64)) < 4e18

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. distribute-frac-neg 28.5%

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

   5. Applied egg-rr 28.5%

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

      1. *-lft-identity 28.5%

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

      2. distribute-neg-frac2 28.5%

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

      3. associate-*l* 29.4%

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

      4. associate-*r/ 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in C around inf 27.5%

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

      1. *-commutative 27.5%

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

   10. Simplified 27.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-258

   1. Initial program 11.1%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. associate-/l* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   5. Simplified 13.3%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 4e18 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e240

   1. Initial program 33.5%

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

   3. Taylor expanded in F around 0 38.0%

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

      1. mul-1-neg 38.0%

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

      2. *-commutative 38.0%

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

      3. distribute-rgt-neg-in 38.0%

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

      4. associate-/l* 47.3%

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

      5. cancel-sign-sub-inv 47.3%

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

      6. metadata-eval 47.3%

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

      7. +-commutative 47.3%

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

   5. Simplified 61.9%

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

      1. fma-undefine 61.9%

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

   7. Applied egg-rr 61.9%

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

   if 2.00000000000000003e240 < (pow.f64 B #s(literal 2 binary64))

   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 B around inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. *-commutative 25.8%

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

      3. distribute-rgt-neg-in 25.8%

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

   5. Simplified 25.8%

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

      1. sqrt-div 38.2%

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

   7. Applied egg-rr 38.2%

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

      1. distribute-rgt-neg-out 38.2%

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

      2. sqrt-undiv 25.8%

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

      3. sqrt-prod 26.0%

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

      4. pow1/2 26.0%

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

      5. neg-sub0 26.0%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.0%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.0%

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

       2. associate-*r/ 26.0%

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

       3. *-commutative 26.0%

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

   11. Simplified 26.0%

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

       1. sqrt-div 38.4%

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

       2. *-commutative 38.4%

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

   13. Applied egg-rr 38.4%

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

       1. *-commutative 38.4%

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

   15. Simplified 38.4%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := {B\_m}^{2} - t\_1\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_3\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{{B\_m}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (* C (* 4.0 A)))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3 (- (pow B_m 2.0) t_1)))
   (if (<= (pow B_m 2.0) 1e-321)
     (/ (sqrt (* (* 2.0 t_3) (* 2.0 (* C F)))) t_2)
     (if (<= (pow B_m 2.0) 4e-233)
       (* (sqrt (* F (/ -0.5 A))) t_0)
       (if (<= (pow B_m 2.0) 2e-95)
         (/ (sqrt (* (* 2.0 (* t_3 F)) (+ C (hypot B_m C)))) t_2)
         (if (<= (pow B_m 2.0) 2e+240)
           (*
            (sqrt
             (*
              F
              (/
               (+ A (+ C (hypot B_m (- A C))))
               (+ (pow B_m 2.0) (* -4.0 (* A C))))))
            t_0)
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = pow(B_m, 2.0) - t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	} else if (pow(B_m, 2.0) <= 4e-233) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 2e-95) {
		tmp = sqrt(((2.0 * (t_3 * F)) * (C + hypot(B_m, C)))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+240) {
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / (pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = t_1 - Math.pow(B_m, 2.0);
	double t_3 = Math.pow(B_m, 2.0) - t_1;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = Math.sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 4e-233) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 2e-95) {
		tmp = Math.sqrt(((2.0 * (t_3 * F)) * (C + Math.hypot(B_m, C)))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+240) {
		tmp = Math.sqrt((F * ((A + (C + Math.hypot(B_m, (A - C)))) / (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = C * (4.0 * A)
	t_2 = t_1 - math.pow(B_m, 2.0)
	t_3 = math.pow(B_m, 2.0) - t_1
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = math.sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2
	elif math.pow(B_m, 2.0) <= 4e-233:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	elif math.pow(B_m, 2.0) <= 2e-95:
		tmp = math.sqrt(((2.0 * (t_3 * F)) * (C + math.hypot(B_m, C)))) / t_2
	elif math.pow(B_m, 2.0) <= 2e+240:
		tmp = math.sqrt((F * ((A + (C + math.hypot(B_m, (A - C)))) / (math.pow(B_m, 2.0) + (-4.0 * (A * C)))))) * t_0
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64((B_m ^ 2.0) - t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_3) * Float64(2.0 * Float64(C * F)))) / t_2);
	elseif ((B_m ^ 2.0) <= 4e-233)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 2e-95)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(C + hypot(B_m, C)))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))))) * t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	t_1 = C * (4.0 * A);
	t_2 = t_1 - (B_m ^ 2.0);
	t_3 = (B_m ^ 2.0) - t_1;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = sqrt(((2.0 * t_3) * (2.0 * (C * F)))) / t_2;
	elseif ((B_m ^ 2.0) <= 4e-233)
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	elseif ((B_m ^ 2.0) <= 2e-95)
		tmp = sqrt(((2.0 * (t_3 * F)) * (C + hypot(B_m, C)))) / t_2;
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / ((B_m ^ 2.0) + (-4.0 * (A * C)))))) * t_0;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], N[(N[Sqrt[N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-233], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-95], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+240], 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[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322

   1. Initial program 24.5%

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

   3. Taylor expanded in A around -inf 30.4%

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

      1. *-un-lft-identity 30.4%

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

      2. distribute-frac-neg 30.4%

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

   5. Applied egg-rr 30.4%

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

      1. *-lft-identity 30.4%

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

      2. distribute-neg-frac2 30.4%

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

      3. associate-*l* 30.5%

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

      4. associate-*r/ 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in C around inf 30.5%

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

      1. *-commutative 30.5%

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

   10. Simplified 30.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999983e-233

   1. Initial program 8.9%

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

   3. Taylor expanded in F around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. *-commutative 9.6%

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

      3. distribute-rgt-neg-in 9.6%

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

      4. associate-/l* 9.6%

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

      5. cancel-sign-sub-inv 9.6%

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

      6. metadata-eval 9.6%

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

      7. +-commutative 9.6%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around -inf 18.4%

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

   if 3.99999999999999983e-233 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999998e-95

   1. Initial program 52.8%

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

   3. Taylor expanded in A around 0 45.1%

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

      1. unpow2 45.1%

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

      2. unpow2 45.1%

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

      3. hypot-define 49.7%

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

   5. Simplified 49.7%

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

   if 1.99999999999999998e-95 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e240

   1. Initial program 28.7%

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

   3. Taylor expanded in F around 0 31.3%

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

      1. mul-1-neg 31.3%

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

      2. *-commutative 31.3%

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

      3. distribute-rgt-neg-in 31.3%

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

      4. associate-/l* 38.4%

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

      5. cancel-sign-sub-inv 38.4%

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

      6. metadata-eval 38.4%

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

      7. +-commutative 38.4%

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

   5. Simplified 52.5%

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

      1. fma-undefine 52.5%

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

   7. Applied egg-rr 52.5%

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

   if 2.00000000000000003e240 < (pow.f64 B #s(literal 2 binary64))

   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 B around inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. *-commutative 25.8%

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

      3. distribute-rgt-neg-in 25.8%

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

   5. Simplified 25.8%

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

      1. sqrt-div 38.2%

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

   7. Applied egg-rr 38.2%

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

      1. distribute-rgt-neg-out 38.2%

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

      2. sqrt-undiv 25.8%

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

      3. sqrt-prod 26.0%

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

      4. pow1/2 26.0%

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

      5. neg-sub0 26.0%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.0%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.0%

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

       2. associate-*r/ 26.0%

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

       3. *-commutative 26.0%

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

   11. Simplified 26.0%

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

       1. sqrt-div 38.4%

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

       2. *-commutative 38.4%

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

   13. Applied egg-rr 38.4%

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

       1. *-commutative 38.4%

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

   15. Simplified 38.4%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := C \cdot \left(4 \cdot A\right)\\ t_2 := \frac{\sqrt{\left(2 \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot t\_0\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B\_m, C\right)}{{B\_m}^{2}}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (t_1 (* C (* 4.0 A)))
        (t_2
         (/
          (sqrt (* (* 2.0 (- (pow B_m 2.0) t_1)) (* 2.0 (* C F))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-321)
     t_2
     (if (<= (pow B_m 2.0) 1e-257)
       (* (sqrt (* F (/ -0.5 A))) t_0)
       (if (<= (pow B_m 2.0) 4e+18)
         t_2
         (if (<= (pow B_m 2.0) 2e+240)
           (* (sqrt (* F (/ (+ C (hypot B_m C)) (pow B_m 2.0)))) t_0)
           (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = sqrt(((2.0 * (pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 1e-257) {
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	} else if (pow(B_m, 2.0) <= 4e+18) {
		tmp = t_2;
	} else if (pow(B_m, 2.0) <= 2e+240) {
		tmp = sqrt((F * ((C + hypot(B_m, C)) / pow(B_m, 2.0)))) * t_0;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double t_1 = C * (4.0 * A);
	double t_2 = Math.sqrt(((2.0 * (Math.pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - Math.pow(B_m, 2.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e-257) {
		tmp = Math.sqrt((F * (-0.5 / A))) * t_0;
	} else if (Math.pow(B_m, 2.0) <= 4e+18) {
		tmp = t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+240) {
		tmp = Math.sqrt((F * ((C + Math.hypot(B_m, C)) / Math.pow(B_m, 2.0)))) * t_0;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	t_1 = C * (4.0 * A)
	t_2 = math.sqrt(((2.0 * (math.pow(B_m, 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - math.pow(B_m, 2.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = t_2
	elif math.pow(B_m, 2.0) <= 1e-257:
		tmp = math.sqrt((F * (-0.5 / A))) * t_0
	elif math.pow(B_m, 2.0) <= 4e+18:
		tmp = t_2
	elif math.pow(B_m, 2.0) <= 2e+240:
		tmp = math.sqrt((F * ((C + math.hypot(B_m, C)) / math.pow(B_m, 2.0)))) * t_0
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(C * Float64(4.0 * A))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64((B_m ^ 2.0) - t_1)) * Float64(2.0 * Float64(C * F)))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * t_0);
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C + hypot(B_m, C)) / (B_m ^ 2.0)))) * t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	t_1 = C * (4.0 * A);
	t_2 = sqrt(((2.0 * ((B_m ^ 2.0) - t_1)) * (2.0 * (C * F)))) / (t_1 - (B_m ^ 2.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = sqrt((F * (-0.5 / A))) * t_0;
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_2;
	elseif ((B_m ^ 2.0) <= 2e+240)
		tmp = sqrt((F * ((C + hypot(B_m, C)) / (B_m ^ 2.0)))) * t_0;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-257], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+18], t$95$2, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+240], N[(N[Sqrt[N[(F * N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322 or 9.9999999999999998e-258 < (pow.f64 B #s(literal 2 binary64)) < 4e18

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. distribute-frac-neg 28.5%

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

   5. Applied egg-rr 28.5%

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

      1. *-lft-identity 28.5%

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

      2. distribute-neg-frac2 28.5%

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

      3. associate-*l* 29.4%

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

      4. associate-*r/ 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in C around inf 27.5%

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

      1. *-commutative 27.5%

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

   10. Simplified 27.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-258

   1. Initial program 11.1%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. associate-/l* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   5. Simplified 13.3%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 4e18 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000003e240

   1. Initial program 33.5%

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

   3. Taylor expanded in F around 0 38.0%

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

      1. mul-1-neg 38.0%

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

      2. *-commutative 38.0%

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

      3. distribute-rgt-neg-in 38.0%

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

      4. associate-/l* 47.3%

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

      5. cancel-sign-sub-inv 47.3%

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

      6. metadata-eval 47.3%

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

      7. +-commutative 47.3%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in A around 0 45.7%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. hypot-define 52.1%

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

   8. Simplified 52.1%

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

   if 2.00000000000000003e240 < (pow.f64 B #s(literal 2 binary64))

   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 B around inf 25.8%

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

      1. mul-1-neg 25.8%

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

      2. *-commutative 25.8%

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

      3. distribute-rgt-neg-in 25.8%

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

   5. Simplified 25.8%

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

      1. sqrt-div 38.2%

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

   7. Applied egg-rr 38.2%

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

      1. distribute-rgt-neg-out 38.2%

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

      2. sqrt-undiv 25.8%

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

      3. sqrt-prod 26.0%

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

      4. pow1/2 26.0%

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

      5. neg-sub0 26.0%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 26.0%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 26.0%

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

       2. associate-*r/ 26.0%

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

       3. *-commutative 26.0%

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

   11. Simplified 26.0%

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

       1. sqrt-div 38.4%

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

       2. *-commutative 38.4%

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

   13. Applied egg-rr 38.4%

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

       1. *-commutative 38.4%

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

   15. Simplified 38.4%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+240}:\\ \;\;\;\;\sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B, C\right)}{{B}^{2}}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A)))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-321)
     t_1
     (if (<= (pow B_m 2.0) 1e-257)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 5e-46)
         t_1
         (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 1e-257) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 5e-46) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    if ((b_m ** 2.0d0) <= 1d-321) then
        tmp = t_1
    else if ((b_m ** 2.0d0) <= 1d-257) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else if ((b_m ** 2.0d0) <= 5d-46) then
        tmp = t_1
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 t_0 = C * (4.0 * A);
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-257) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 5e-46) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = t_1
	elif math.pow(B_m, 2.0) <= 1e-257:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 5e-46:
		tmp = t_1
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = 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)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 5e-46)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 5e-46)
		tmp = t_1;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-257], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-46], t$95$1, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322 or 9.9999999999999998e-258 < (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999992e-46

   1. Initial program 28.5%

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

   3. Taylor expanded in A around -inf 28.3%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-258

   1. Initial program 11.1%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. associate-/l* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   5. Simplified 13.3%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 4.99999999999999992e-46 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in B around inf 22.8%

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

      1. mul-1-neg 22.8%

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

      2. *-commutative 22.8%

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

      3. distribute-rgt-neg-in 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-div 29.8%

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

   7. Applied egg-rr 29.8%

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

      1. distribute-rgt-neg-out 29.8%

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

      2. sqrt-undiv 22.8%

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

      3. sqrt-prod 22.9%

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

      4. pow1/2 22.9%

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

      5. neg-sub0 22.9%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 22.9%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 22.9%

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

       2. associate-*r/ 22.9%

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

       3. *-commutative 22.9%

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

   11. Simplified 22.9%

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

       1. sqrt-div 29.9%

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

       2. *-commutative 29.9%

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

   13. Applied egg-rr 29.9%

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

       1. *-commutative 29.9%

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

   15. Simplified 29.9%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(4 \cdot A\right)\\ t_1 := \frac{\sqrt{\left(2 \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-321}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* 4.0 A)))
        (t_1
         (/
          (sqrt (* (* 2.0 (- (pow B_m 2.0) t_0)) (* 2.0 (* C F))))
          (- t_0 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-321)
     t_1
     (if (<= (pow B_m 2.0) 1e-257)
       (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 4e+18)
         t_1
         (/ (sqrt (* 2.0 F)) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (4.0 * A);
	double t_1 = sqrt(((2.0 * (pow(B_m, 2.0) - t_0)) * (2.0 * (C * F)))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-321) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 1e-257) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 4e+18) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (4.0d0 * a)
    t_1 = sqrt(((2.0d0 * ((b_m ** 2.0d0) - t_0)) * (2.0d0 * (c * f)))) / (t_0 - (b_m ** 2.0d0))
    if ((b_m ** 2.0d0) <= 1d-321) then
        tmp = t_1
    else if ((b_m ** 2.0d0) <= 1d-257) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else if ((b_m ** 2.0d0) <= 4d+18) then
        tmp = t_1
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 t_0 = C * (4.0 * A);
	double t_1 = Math.sqrt(((2.0 * (Math.pow(B_m, 2.0) - t_0)) * (2.0 * (C * F)))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-321) {
		tmp = t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-257) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 4e+18) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = C * (4.0 * A)
	t_1 = math.sqrt(((2.0 * (math.pow(B_m, 2.0) - t_0)) * (2.0 * (C * F)))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-321:
		tmp = t_1
	elif math.pow(B_m, 2.0) <= 1e-257:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 4e+18:
		tmp = t_1
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(4.0 * A))
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64((B_m ^ 2.0) - t_0)) * Float64(2.0 * Float64(C * F)))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = C * (4.0 * A);
	t_1 = sqrt(((2.0 * ((B_m ^ 2.0) - t_0)) * (2.0 * (C * F)))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-321)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 1e-257)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 4e+18)
		tmp = t_1;
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-321], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-257], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+18], t$95$1, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.98013e-322 or 9.9999999999999998e-258 < (pow.f64 B #s(literal 2 binary64)) < 4e18

   1. Initial program 27.8%

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

   3. Taylor expanded in A around -inf 28.5%

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

      1. *-un-lft-identity 28.5%

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

      2. distribute-frac-neg 28.5%

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

   5. Applied egg-rr 28.5%

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

      1. *-lft-identity 28.5%

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

      2. distribute-neg-frac2 28.5%

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

      3. associate-*l* 29.4%

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

      4. associate-*r/ 29.4%

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

   7. Simplified 29.4%

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

   8. Taylor expanded in C around inf 27.5%

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

      1. *-commutative 27.5%

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

   10. Simplified 27.5%

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

   if 9.98013e-322 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-258

   1. Initial program 11.1%

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

   3. Taylor expanded in F around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. associate-/l* 5.8%

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

      5. cancel-sign-sub-inv 5.8%

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

      6. metadata-eval 5.8%

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

      7. +-commutative 5.8%

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

   5. Simplified 13.3%

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

   6. Taylor expanded in A around -inf 20.6%

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

   if 4e18 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 24.4%

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

      1. mul-1-neg 24.4%

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

      2. *-commutative 24.4%

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

      3. distribute-rgt-neg-in 24.4%

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

   5. Simplified 24.4%

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

      1. sqrt-div 32.3%

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

   7. Applied egg-rr 32.3%

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

      1. distribute-rgt-neg-out 32.3%

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

      2. sqrt-undiv 24.4%

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

      3. sqrt-prod 24.6%

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

      4. pow1/2 24.6%

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

      5. neg-sub0 24.6%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 24.6%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 24.6%

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

       2. associate-*r/ 24.6%

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

       3. *-commutative 24.6%

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

   11. Simplified 24.6%

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

       1. sqrt-div 32.4%

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

       2. *-commutative 32.4%

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

   13. Applied egg-rr 32.4%

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

       1. *-commutative 32.4%

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

   15. Simplified 32.4%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-321}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-257}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \left(2 \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(4 \cdot A\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.16 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.16e+54)
   (* (sqrt (* F (/ -0.5 A))) (- (sqrt 2.0)))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.16e+54) {
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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 (b_m <= 1.16d+54) then
        tmp = sqrt((f * ((-0.5d0) / a))) * -sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 (B_m <= 1.16e+54) {
		tmp = Math.sqrt((F * (-0.5 / A))) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.16e+54:
		tmp = math.sqrt((F * (-0.5 / A))) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.16e+54)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / A))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.16e+54)
		tmp = sqrt((F * (-0.5 / A))) * -sqrt(2.0);
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.16e+54], N[(N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.1600000000000001e54

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0 17.5%

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

      1. mul-1-neg 17.5%

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

      2. *-commutative 17.5%

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

      3. distribute-rgt-neg-in 17.5%

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

      4. associate-/l* 19.8%

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

      5. cancel-sign-sub-inv 19.8%

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

      6. metadata-eval 19.8%

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

      7. +-commutative 19.8%

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

   5. Simplified 28.0%

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

   6. Taylor expanded in A around -inf 16.5%

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

   if 1.1600000000000001e54 < 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 B around inf 50.9%

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

      1. mul-1-neg 50.9%

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

      2. *-commutative 50.9%

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

      3. distribute-rgt-neg-in 50.9%

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

   5. Simplified 50.9%

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

      1. sqrt-div 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. distribute-rgt-neg-out 71.0%

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

      2. sqrt-undiv 50.9%

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

      3. sqrt-prod 51.2%

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

      4. pow1/2 51.2%

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

      5. neg-sub0 51.2%

         \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

      6. pow1/2 51.2%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   9. Applied egg-rr 51.2%

      \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   10. Step-by-step derivation

       1. neg-sub0 51.2%

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

       2. associate-*r/ 51.3%

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

       3. *-commutative 51.3%

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

   11. Simplified 51.3%

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

       1. sqrt-div 71.3%

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

       2. *-commutative 71.3%

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

   13. Applied egg-rr 71.3%

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

       1. *-commutative 71.3%

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

   15. Simplified 71.3%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.16 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt (* 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) {
	return sqrt((2.0 * F)) * -sqrt((1.0 / 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((2.0d0 * f)) * -sqrt((1.0d0 / 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((2.0 * F)) * -Math.sqrt((1.0 / 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((2.0 * F)) * -math.sqrt((1.0 / 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(2.0 * F)) * Float64(-sqrt(Float64(1.0 / 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((2.0 * F)) * -sqrt((1.0 / 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[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

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

   1. sqrt-div 17.6%

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

7. Applied egg-rr 17.6%

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

   1. distribute-rgt-neg-out 17.6%

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

   2. sqrt-undiv 13.7%

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

   3. sqrt-prod 13.8%

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

   4. pow1/2 13.9%

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

   5. neg-sub0 13.9%

      \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   6. pow1/2 13.8%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
10. Step-by-step derivation

    1. neg-sub0 13.8%

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

    2. associate-*r/ 13.8%

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

    3. *-commutative 13.8%

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

11. Simplified 13.8%

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

    1. pow1/2 14.0%

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

    2. div-inv 14.0%

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

    3. unpow-prod-down 17.6%

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

    4. pow1/2 17.6%

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

    5. *-commutative 17.6%

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

13. Applied egg-rr 17.6%

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

    1. *-commutative 17.6%

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

    2. unpow1/2 17.6%

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

15. Simplified 17.6%

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

16. Final simplification 17.6%

    \[\leadsto \sqrt{2 \cdot F} \cdot \left(-\sqrt{\frac{1}{B}}\right) \]
17. Add Preprocessing

Alternative 13: 27.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. distribute-rgt-neg-out 13.7%

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

   3. pow1/2 13.7%

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

   4. pow1/2 13.9%

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

   5. pow-prod-down 13.9%

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

7. Applied egg-rr 13.9%

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

   1. unpow1 13.9%

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

   2. unpow1/2 13.8%

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

9. Simplified 13.8%

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

    1. add-sqr-sqrt 13.8%

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

    2. pow1/2 13.8%

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

    3. pow1/2 13.9%

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

    4. pow-prod-down 19.6%

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

    5. pow2 19.6%

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

11. Applied egg-rr 19.6%

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

    1. unpow1/2 19.6%

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

    2. unpow2 19.6%

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

    3. rem-sqrt-square 28.0%

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

    4. associate-*r/ 28.0%

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

    5. *-commutative 28.0%

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

13. Simplified 28.0%

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

14. Final simplification 28.0%

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

Alternative 14: 35.5% accurate, 3.1× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt 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) {
	return sqrt(F) * -sqrt((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) * -sqrt((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) * -Math.sqrt((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) * -math.sqrt((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(F) * Float64(-sqrt(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) * -sqrt((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[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

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

   1. sqrt-div 17.6%

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

7. Applied egg-rr 17.6%

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

   1. distribute-rgt-neg-out 17.6%

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

   2. sqrt-undiv 13.7%

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

   3. sqrt-prod 13.8%

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

   4. pow1/2 13.9%

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

   5. neg-sub0 13.9%

      \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   6. pow1/2 13.8%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
10. Step-by-step derivation

    1. neg-sub0 13.8%

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

    2. associate-*r/ 13.8%

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

    3. *-commutative 13.8%

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

11. Simplified 13.8%

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

    1. pow1/2 14.0%

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

    2. associate-/l* 14.0%

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

    3. unpow-prod-down 17.6%

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

    4. pow1/2 17.6%

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

13. Applied egg-rr 17.6%

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

    1. unpow1/2 17.6%

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

15. Simplified 17.6%

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

16. Final simplification 17.6%

    \[\leadsto \sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right) \]
17. Add Preprocessing

Alternative 15: 35.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

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

   1. sqrt-div 17.6%

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

7. Applied egg-rr 17.6%

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

   1. distribute-rgt-neg-out 17.6%

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

   2. sqrt-undiv 13.7%

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

   3. sqrt-prod 13.8%

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

   4. pow1/2 13.9%

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

   5. neg-sub0 13.9%

      \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   6. pow1/2 13.8%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
10. Step-by-step derivation

    1. neg-sub0 13.8%

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

    2. associate-*r/ 13.8%

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

    3. *-commutative 13.8%

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

11. Simplified 13.8%

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

    1. sqrt-div 17.6%

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

    2. *-commutative 17.6%

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

13. Applied egg-rr 17.6%

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

    1. *-commutative 17.6%

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

15. Simplified 17.6%

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

16. Final simplification 17.6%

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

Alternative 16: 27.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. distribute-rgt-neg-out 13.7%

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

   3. pow1/2 13.7%

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

   4. pow1/2 13.9%

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

   5. pow-prod-down 13.9%

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

7. Applied egg-rr 13.9%

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

   1. unpow1 13.9%

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

   2. unpow1/2 13.8%

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

9. Simplified 13.8%

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

10. Final simplification 13.8%

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

Alternative 17: 27.0% accurate, 6.0× speedup

Math

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

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 20.0%

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

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

   2. *-commutative 13.7%

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

   3. distribute-rgt-neg-in 13.7%

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

5. Simplified 13.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]
6. Step-by-step derivation

   1. sqrt-div 17.6%

      \[\leadsto \sqrt{2} \cdot \left(-\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}}\right) \]

7. Applied egg-rr 17.6%

   \[\leadsto \sqrt{2} \cdot \left(-\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}}\right) \]
8. Step-by-step derivation

   1. distribute-rgt-neg-out 17.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \frac{\sqrt{F}}{\sqrt{B}}} \]

   2. sqrt-undiv 13.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   3. sqrt-prod 13.8%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   4. pow1/2 13.9%

      \[\leadsto -\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   5. neg-sub0 13.9%

      \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   6. pow1/2 13.8%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
10. Step-by-step derivation

    1. neg-sub0 13.8%

       \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]

    2. associate-*r/ 13.8%

       \[\leadsto -\sqrt{\color{blue}{\frac{2 \cdot F}{B}}} \]

    3. *-commutative 13.8%

       \[\leadsto -\sqrt{\frac{\color{blue}{F \cdot 2}}{B}} \]

11. Simplified 13.8%

    \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]
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 18: 27.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

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 20.0%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. 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%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   2. *-commutative 13.7%

      \[\leadsto -\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   3. distribute-rgt-neg-in 13.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]
6. Step-by-step derivation

   1. sqrt-div 17.6%

      \[\leadsto \sqrt{2} \cdot \left(-\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}}\right) \]

7. Applied egg-rr 17.6%

   \[\leadsto \sqrt{2} \cdot \left(-\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}}\right) \]
8. Step-by-step derivation

   1. distribute-rgt-neg-out 17.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \frac{\sqrt{F}}{\sqrt{B}}} \]

   2. sqrt-undiv 13.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   3. sqrt-prod 13.8%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

   4. pow1/2 13.9%

      \[\leadsto -\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   5. neg-sub0 13.9%

      \[\leadsto \color{blue}{0 - {\left(2 \cdot \frac{F}{B}\right)}^{0.5}} \]

   6. pow1/2 13.8%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
10. Step-by-step derivation

    1. neg-sub0 13.8%

       \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]

    2. associate-*r/ 13.8%

       \[\leadsto -\sqrt{\color{blue}{\frac{2 \cdot F}{B}}} \]

    3. *-commutative 13.8%

       \[\leadsto -\sqrt{\frac{\color{blue}{F \cdot 2}}{B}} \]

11. Simplified 13.8%

    \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot 2}{B}}} \]

12. Final simplification 13.8%

    \[\leadsto -\sqrt{\frac{2 \cdot F}{B}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))