ABCF->ab-angle a

Percentage Accurate: 19.0% → 53.5%

Time: 19.2s

Alternatives: 13

Speedup: 16.9×

• could not determine a ground truth

  1. A = -2.542687091886861e-184
  2. B = -1.2447986563389958e-57
  3. C = -3.546907088031563e+69
  4. F = 1.1796642671782842e-153

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 53.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 4.75 \cdot 10^{+123}:\\ \;\;\;\;\frac{t\_2}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_2 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (sqrt (* 2.0 F))))
   (if (<= B_m 4.2e-87)
     (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C))) t_1)
     (if (<= B_m 4.8e-7)
       (/
        (*
         (sqrt (* 2.0 (fma 2.0 C (/ (* B_m (* B_m -0.5)) A))))
         (sqrt (* F (fma B_m B_m (* -4.0 (* A C))))))
        t_1)
       (if (<= B_m 4.75e+123)
         (* (/ t_2 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
         (- (* t_2 (sqrt (/ 1.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 4.2e-87) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 4.8e-7) {
		tmp = (sqrt((2.0 * fma(2.0, C, ((B_m * (B_m * -0.5)) / A)))) * sqrt((F * fma(B_m, B_m, (-4.0 * (A * C)))))) / t_1;
	} else if (B_m <= 4.75e+123) {
		tmp = (t_2 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_2 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 4.2e-87)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 4.8e-7)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)))) * sqrt(Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) / t_1);
	elseif (B_m <= 4.75e+123)
		tmp = Float64(Float64(t_2 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_2 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 4.2e-87], 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] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 4.8e-7], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 4.75e+123], N[(N[(t$95$2 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$2 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.20000000000000014e-87

   1. Initial program 19.6%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 15.5

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

   5. Simplified 15.5%

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

   if 4.20000000000000014e-87 < B < 4.79999999999999957e-7

   1. Initial program 28.0%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 34.6

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

   5. Simplified 34.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 29.3%

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

   if 4.79999999999999957e-7 < B < 4.7499999999999998e123

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 4.7499999999999998e123 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 4.75 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := \mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right)\\ t_3 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 6.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_2 \cdot t\_1\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{t\_2} \cdot \sqrt{2 \cdot t\_1}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 5.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_3}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_3 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 (* -4.0 (* A C))))
        (t_1 (* F t_0))
        (t_2 (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)))
        (t_3 (sqrt (* 2.0 F))))
   (if (<= B_m 6.8e-199)
     (/ (sqrt (* 2.0 (* t_2 t_1))) (- t_0))
     (if (<= B_m 4.7e-7)
       (/ (* (sqrt t_2) (sqrt (* 2.0 t_1))) (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= B_m 5.2e+122)
         (* (/ t_3 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
         (- (* t_3 (sqrt (/ 1.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = F * t_0;
	double t_2 = fma(2.0, C, ((B_m * (B_m * -0.5)) / A));
	double t_3 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 6.8e-199) {
		tmp = sqrt((2.0 * (t_2 * t_1))) / -t_0;
	} else if (B_m <= 4.7e-7) {
		tmp = (sqrt(t_2) * sqrt((2.0 * t_1))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (B_m <= 5.2e+122) {
		tmp = (t_3 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_3 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(F * t_0)
	t_2 = fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A))
	t_3 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 6.8e-199)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_2 * t_1))) / Float64(-t_0));
	elseif (B_m <= 4.7e-7)
		tmp = Float64(Float64(sqrt(t_2) * sqrt(Float64(2.0 * t_1))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif (B_m <= 5.2e+122)
		tmp = Float64(Float64(t_3 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_3 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 6.8e-199], N[(N[Sqrt[N[(2.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 4.7e-7], N[(N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.2e+122], N[(N[(t$95$3 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$3 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 6.80000000000000011e-199

   1. Initial program 18.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 16.4

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

   5. Simplified 16.4%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 16.4%

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

   if 6.80000000000000011e-199 < B < 4.7e-7

   1. Initial program 26.6%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 20.7

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

   5. Simplified 20.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. *-lowering-*.f64 N/A

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

      5. pow1/2 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. pow1/2 N/A

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

   7. Applied egg-rr 24.2%

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

   if 4.7e-7 < B < 5.20000000000000015e122

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 5.20000000000000015e122 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := \mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right)\\ t_3 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 2.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t\_2 \cdot t\_1\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_2} \cdot \sqrt{t\_1}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 9.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_3}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_3 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 (* -4.0 (* A C))))
        (t_1 (* F t_0))
        (t_2 (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)))
        (t_3 (sqrt (* 2.0 F))))
   (if (<= B_m 2.2e-202)
     (/ (sqrt (* 2.0 (* t_2 t_1))) (- t_0))
     (if (<= B_m 4.5e-7)
       (/ (* (sqrt (* 2.0 t_2)) (sqrt t_1)) (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= B_m 9.5e+122)
         (* (/ t_3 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
         (- (* t_3 (sqrt (/ 1.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = F * t_0;
	double t_2 = fma(2.0, C, ((B_m * (B_m * -0.5)) / A));
	double t_3 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 2.2e-202) {
		tmp = sqrt((2.0 * (t_2 * t_1))) / -t_0;
	} else if (B_m <= 4.5e-7) {
		tmp = (sqrt((2.0 * t_2)) * sqrt(t_1)) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (B_m <= 9.5e+122) {
		tmp = (t_3 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_3 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(F * t_0)
	t_2 = fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A))
	t_3 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 2.2e-202)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_2 * t_1))) / Float64(-t_0));
	elseif (B_m <= 4.5e-7)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_2)) * sqrt(t_1)) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif (B_m <= 9.5e+122)
		tmp = Float64(Float64(t_3 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_3 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 2.2e-202], N[(N[Sqrt[N[(2.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 4.5e-7], N[(N[(N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 9.5e+122], N[(N[(t$95$3 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$3 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.20000000000000008e-202

   1. Initial program 18.3%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 16.5

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

   5. Simplified 16.5%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 16.5%

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

   if 2.20000000000000008e-202 < B < 4.4999999999999998e-7

   1. Initial program 28.2%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 20.3

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

   5. Simplified 20.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sqrt-prod N/A

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

      10. pow1/2 N/A

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

      11. *-lowering-*.f64 N/A

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

   7. Applied egg-rr 23.7%

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

   if 4.4999999999999998e-7 < B < 9.49999999999999986e122

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 9.49999999999999986e122 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-13)
   (* (sqrt (* F (/ (* -0.5 (* B_m B_m)) A))) (- (/ (sqrt 2.0) B_m)))
   (/ (- (sqrt F)) (sqrt (* B_m 0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-13) {
		tmp = sqrt((F * ((-0.5 * (B_m * B_m)) / A))) * -(sqrt(2.0) / B_m);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 2d-13) then
        tmp = sqrt((f * (((-0.5d0) * (b_m * b_m)) / a))) * -(sqrt(2.0d0) / b_m)
    else
        tmp = -sqrt(f) / sqrt((b_m * 0.5d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-13) {
		tmp = Math.sqrt((F * ((-0.5 * (B_m * B_m)) / A))) * -(Math.sqrt(2.0) / B_m);
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-13:
		tmp = math.sqrt((F * ((-0.5 * (B_m * B_m)) / A))) * -(math.sqrt(2.0) / B_m)
	else:
		tmp = -math.sqrt(F) / math.sqrt((B_m * 0.5))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-13)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-13)
		tmp = sqrt((F * ((-0.5 * (B_m * B_m)) / A))) * -(sqrt(2.0) / B_m);
	else
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-13], N[(N[Sqrt[N[(F * N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-13

   1. Initial program 24.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 6.7

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

   5. Simplified 6.7%

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

   6. Taylor expanded in A around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 7.2

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

   8. Simplified 7.2%

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

   if 2.0000000000000001e-13 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.9%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 19.2

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

   5. Simplified 19.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 24.2

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

   7. Applied egg-rr 24.2%

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

      1. un-div-inv N/A

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

      2. sqrt-div N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. sqrt-prod N/A

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

      6. pow1/2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. /-lowering-/.f64 24.2

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

   9. Applied egg-rr 24.2%

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

       1. distribute-lft-neg-in N/A

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

       2. clear-num N/A

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

       3. sqrt-div N/A

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

       4. metadata-eval N/A

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

       5. un-div-inv N/A

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

       6. /-lowering-/.f64 N/A

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

       7. neg-lowering-neg.f64 N/A

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

       8. sqrt-lowering-sqrt.f64 N/A

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

       9. sqrt-lowering-sqrt.f64 N/A

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

       10. div-inv N/A

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

       11. metadata-eval N/A

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

       12. *-lowering-*.f64 24.2

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

   11. Applied egg-rr 24.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \left(B \cdot B\right)}{A}} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_1\right)\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 3.15 \cdot 10^{+123}:\\ \;\;\;\;\frac{t\_0}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_0 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 F))) (t_1 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= B_m 4.5e-7)
     (/
      (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_1))))
      (- t_1))
     (if (<= B_m 3.15e+123)
       (* (/ t_0 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
       (- (* t_0 (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((2.0 * F));
	double t_1 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (B_m <= 4.5e-7) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_1)))) / -t_1;
	} else if (B_m <= 3.15e+123) {
		tmp = (t_0 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_0 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (B_m <= 4.5e-7)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_1)))) / Float64(-t_1));
	elseif (B_m <= 3.15e+123)
		tmp = Float64(Float64(t_0 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_0 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.5e-7], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 3.15e+123], N[(N[(t$95$0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.4999999999999998e-7

   1. Initial program 20.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 17.3

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

   5. Simplified 17.3%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 17.3%

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

   if 4.4999999999999998e-7 < B < 3.1500000000000001e123

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 3.1500000000000001e123 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_0}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_0 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 F))))
   (if (<= B_m 4.4e-7)
     (*
      (sqrt
       (/
        (* F (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
        (fma B_m B_m (* -4.0 (* A C)))))
      (- (sqrt 2.0)))
     (if (<= B_m 3.1e+122)
       (* (/ t_0 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
       (- (* t_0 (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 4.4e-7) {
		tmp = sqrt(((F * fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) / fma(B_m, B_m, (-4.0 * (A * C))))) * -sqrt(2.0);
	} else if (B_m <= 3.1e+122) {
		tmp = (t_0 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_0 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 4.4e-7)
		tmp = Float64(sqrt(Float64(Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 3.1e+122)
		tmp = Float64(Float64(t_0 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_0 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 4.4e-7], N[(N[Sqrt[N[(N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.1e+122], N[(N[(t$95$0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.4000000000000002e-7

   1. Initial program 20.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 17.3

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

   5. Simplified 17.3%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

   8. Simplified 11.9%

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

   if 4.4000000000000002e-7 < B < 3.09999999999999999e122

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 3.09999999999999999e122 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 4.65 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(\left(2 \cdot C\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.75 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_1}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_1 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 -4.0 (* A C) (* B_m B_m))) (t_1 (sqrt (* 2.0 F))))
   (if (<= B_m 4.65e-7)
     (* (sqrt 2.0) (/ (sqrt (* F (* (* 2.0 C) t_0))) (- t_0)))
     (if (<= B_m 1.75e+122)
       (* (/ t_1 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
       (- (* t_1 (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 4.65e-7) {
		tmp = sqrt(2.0) * (sqrt((F * ((2.0 * C) * t_0))) / -t_0);
	} else if (B_m <= 1.75e+122) {
		tmp = (t_1 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_1 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 4.65e-7)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(Float64(2.0 * C) * t_0))) / Float64(-t_0)));
	elseif (B_m <= 1.75e+122)
		tmp = Float64(Float64(t_1 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_1 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 4.65e-7], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(N[(2.0 * C), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.75e+122], N[(N[(t$95$1 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$1 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.6499999999999999e-7

   1. Initial program 20.5%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 18.9%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 13.8

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

   7. Simplified 13.8%

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

   if 4.6499999999999999e-7 < B < 1.75000000000000007e122

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 1.75000000000000007e122 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.65 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(\left(2 \cdot C\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ \mathbf{if}\;B\_m \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \left(B\_m \cdot B\_m\right)}{A}} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{elif}\;B\_m \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{t\_0}{-B\_m} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-t\_0 \cdot \sqrt{\frac{1}{B\_m}}\\ \end{array} \end{array} \]

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 F))))
   (if (<= B_m 4.3e-7)
     (* (sqrt (* F (/ (* -0.5 (* B_m B_m)) A))) (- (/ (sqrt 2.0) B_m)))
     (if (<= B_m 7.8e+121)
       (* (/ t_0 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))
       (- (* t_0 (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((2.0 * F));
	double tmp;
	if (B_m <= 4.3e-7) {
		tmp = sqrt((F * ((-0.5 * (B_m * B_m)) / A))) * -(sqrt(2.0) / B_m);
	} else if (B_m <= 7.8e+121) {
		tmp = (t_0 / -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))));
	} else {
		tmp = -(t_0 * sqrt((1.0 / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (B_m <= 4.3e-7)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(-0.5 * Float64(B_m * B_m)) / A))) * Float64(-Float64(sqrt(2.0) / B_m)));
	elseif (B_m <= 7.8e+121)
		tmp = Float64(Float64(t_0 / Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))));
	else
		tmp = Float64(-Float64(t_0 * sqrt(Float64(1.0 / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 4.3e-7], N[(N[Sqrt[N[(F * N[(N[(-0.5 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 7.8e+121], N[(N[(t$95$0 / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.3000000000000001e-7

   1. Initial program 20.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 4.6

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

   5. Simplified 4.6%

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

   6. Taylor expanded in A around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 6.2

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

   8. Simplified 6.2%

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

   if 4.3000000000000001e-7 < B < 7.79999999999999967e121

   1. Initial program 39.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.3%

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

   5. Taylor expanded in A around 0

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

      1. *-lowering-*.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-lowering-+.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.1

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

   7. Simplified 43.1%

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

      1. associate-*r* N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 43.1%

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

   if 7.79999999999999967e121 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 47.2

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

   5. Simplified 47.2%

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

      1. sqrt-unprod N/A

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

      2. pow1/2 N/A

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

      3. div-inv N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-down N/A

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

      7. pow1/2 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. pow1/2 N/A

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

      10. sqrt-unprod N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. metadata-eval N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 N/A

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

      20. metadata-eval N/A

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

      21. sqrt-lowering-sqrt.f64 68.4

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

   7. Applied egg-rr 68.4%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 68.6

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

   9. Applied egg-rr 68.6%

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

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5 \cdot \left(B \cdot B\right)}{A}} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-B} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot F} \cdot \sqrt{\frac{1}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.9% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. /-lowering-/.f64 12.5

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

5. Simplified 12.5%

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

   1. sqrt-unprod N/A

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

   2. pow1/2 N/A

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

   3. div-inv N/A

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

   4. associate-*r* N/A

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

   5. unpow-prod-down N/A

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

   6. pow-prod-down N/A

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

   7. pow1/2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. sqrt-unprod N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. pow1/2 N/A

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

   14. metadata-eval N/A

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

   15. sqrt-div N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f64 N/A

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

   20. metadata-eval N/A

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

   21. sqrt-lowering-sqrt.f64 15.8

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

7. Applied egg-rr 15.8%

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

   1. un-div-inv N/A

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

   2. sqrt-div N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. sqrt-prod N/A

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

   6. pow1/2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. /-lowering-/.f64 15.9

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

9. Applied egg-rr 15.9%

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

    1. distribute-lft-neg-in N/A

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

    2. clear-num N/A

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

    3. sqrt-div N/A

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

    4. metadata-eval N/A

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

    5. un-div-inv N/A

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

    6. /-lowering-/.f64 N/A

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

    7. neg-lowering-neg.f64 N/A

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

    8. sqrt-lowering-sqrt.f64 N/A

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

    9. sqrt-lowering-sqrt.f64 N/A

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

    10. div-inv N/A

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

    11. metadata-eval N/A

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

    12. *-lowering-*.f64 15.9

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

11. Applied egg-rr 15.9%

    \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
12. Add Preprocessing

Alternative 10: 35.9% accurate, 12.6× speedup

Math

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

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)) (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(Float64(-sqrt(F)) * 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 19.7%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. /-lowering-/.f64 12.5

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

5. Simplified 12.5%

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

   1. sqrt-unprod N/A

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

   2. pow1/2 N/A

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

   3. div-inv N/A

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

   4. associate-*r* N/A

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

   5. unpow-prod-down N/A

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

   6. pow-prod-down N/A

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

   7. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\sqrt{2}} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\sqrt{2} \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

   9. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{F}}\right) \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   10. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   13. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

   15. sqrt-div N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   21. sqrt-lowering-sqrt.f64 15.8

       \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

7. Applied egg-rr 15.8%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2 \cdot F}}{\sqrt{B}}}\right) \]

   2. sqrt-div N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2 \cdot F}{B}}}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   5. sqrt-prod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \sqrt{\frac{2}{B}}}\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   11. /-lowering-/.f64 15.9

       \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 15.9%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

10. Final simplification 15.9%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 11: 28.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.1 \cdot 10^{+239}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3.1e+239)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* (sqrt (* C 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) {
	double tmp;
	if (C <= 3.1e+239) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = sqrt((C * F)) * (-2.0 / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3.1d+239) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = sqrt((c * f)) * ((-2.0d0) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.1e+239) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = Math.sqrt((C * F)) * (-2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3.1e+239:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = math.sqrt((C * F)) * (-2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 3.1e+239)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(sqrt(Float64(C * F)) * Float64(-2.0 / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 3.1e+239)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = sqrt((C * F)) * (-2.0 / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 3.1e+239], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.10000000000000001e239

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 13.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 13.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      6. *-lowering-*.f64 13.3

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   7. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]

   if 3.10000000000000001e239 < C

   1. Initial program 1.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{{\color{blue}{\left(2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{2}^{\frac{1}{2}} \cdot {\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{2}^{\frac{1}{2}} \cdot {\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{2}^{\frac{1}{2}}}{-1} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   4. Applied egg-rr 1.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\color{blue}{\frac{1}{B}} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \color{blue}{\sqrt{{B}^{2} + {C}^{2}}}\right)}\right) \]

      7. unpow2 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {C}^{2}\right)}}\right)}\right) \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)}\right) \]

      10. *-lowering-*.f64 1.1

          \[\leadsto \frac{\sqrt{2}}{-1} \cdot \left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)}\right) \]

   7. Simplified 1.1%

      \[\leadsto \frac{\sqrt{2}}{-1} \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\right)} \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{C \cdot F} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sqrt{C \cdot F} \cdot \left(\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{C \cdot F} \cdot \color{blue}{\left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{C \cdot F} \cdot \left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{C \cdot F}} \cdot \left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{F \cdot C}} \cdot \left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{F \cdot C}} \cdot \left(-1 \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{F \cdot C} \cdot \left(-1 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B}\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \sqrt{F \cdot C} \cdot \left(-1 \cdot \frac{\color{blue}{2}}{B}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \sqrt{F \cdot C} \cdot \color{blue}{\frac{-1 \cdot 2}{B}} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{F \cdot C} \cdot \frac{\color{blue}{-2}}{B} \]

      13. /-lowering-/.f64 6.7

          \[\leadsto \sqrt{F \cdot C} \cdot \color{blue}{\frac{-2}{B}} \]

   10. Simplified 6.7%

       \[\leadsto \color{blue}{\sqrt{F \cdot C} \cdot \frac{-2}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.1 \cdot 10^{+239}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.8% accurate, 16.9× 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 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. /-lowering-/.f64 12.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 12.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. *-lowering-*.f64 12.6

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 12.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Add Preprocessing

Alternative 13: 27.8% accurate, 16.9× 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 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. /-lowering-/.f64 12.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 12.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. *-lowering-*.f64 12.6

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 12.6%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   4. /-lowering-/.f64 12.6

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 12.6%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))