ABCF->ab-angle a

Percentage Accurate: 18.6% → 60.6%

Time: 22.0s

Alternatives: 22

Speedup: 16.9×

• could not determine a ground truth

  1. A = -2.2893588705227514e+132
  2. B = -4.046708918209794e-42
  3. C = 1.088847118188673e-215
  4. F = 1.0981683382058174e+211

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.6% 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: 60.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \sqrt{2 \cdot F}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{t\_0} \cdot t\_4\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}} \cdot \left(t\_4 \cdot \sqrt{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_0\right)\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (sqrt (* 2.0 F))))
   (if (<= t_3 (- INFINITY))
     (/
      (* (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))) (* (sqrt t_0) t_4))
      t_2)
     (if (<= t_3 -2e-195)
       (/
        (*
         (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))
         (* t_4 (sqrt (fma B_m B_m (* -4.0 (* A C))))))
        t_2)
       (if (<= t_3 5e+61)
         (/
          -1.0
          (/
           t_0
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_0))))))
         (if (<= t_3 INFINITY)
           (/
            (*
             (sqrt (* 2.0 C))
             (sqrt (* 2.0 (* F (fma -4.0 (* A C) (* B_m B_m))))))
            t_2)
           (/ (- (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 t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = sqrt((2.0 * F));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) * (sqrt(t_0) * t_4)) / t_2;
	} else if (t_3 <= -2e-195) {
		tmp = (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) * (t_4 * sqrt(fma(B_m, B_m, (-4.0 * (A * C)))))) / t_2;
	} else if (t_3 <= 5e+61) {
		tmp = -1.0 / (t_0 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) * sqrt((2.0 * (F * fma(-4.0, (A * C), (B_m * B_m)))))) / t_2;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) * Float64(sqrt(t_0) * t_4)) / t_2);
	elseif (t_3 <= -2e-195)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) * Float64(t_4 * sqrt(fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) / t_2);
	elseif (t_3 <= 5e+61)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_0))))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * sqrt(Float64(2.0 * Float64(F * fma(-4.0, Float64(A * C), Float64(B_m * B_m)))))) / t_2);
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-195], N[(N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 * N[Sqrt[N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 5e+61], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 9.1%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 21.0

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

   7. Simplified 21.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      13. pow1/2 N/A

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

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

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

      15. *-lowering-*.f64 27.0

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

   9. Applied egg-rr 27.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 96.6%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-down N/A

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

      11. pow1/2 N/A

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

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

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

   6. Applied egg-rr 97.6%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000018e61

   1. Initial program 16.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.9

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

   7. Simplified 18.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 26.3%

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

   if 5.00000000000000018e61 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 30.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 36.3

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

   7. Simplified 36.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{2 \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}} \cdot \left(\sqrt{2 \cdot F} \cdot \sqrt{\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}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{t\_0} \cdot \sqrt{2 \cdot F}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}} \cdot \sqrt{F}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_0\right)\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/
      (*
       (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
       (* (sqrt t_0) (sqrt (* 2.0 F))))
      t_2)
     (if (<= t_3 -2e-195)
       (/
        (*
         (sqrt (* 2.0 (fma B_m B_m (* -4.0 (* A C)))))
         (*
          (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))
          (sqrt F)))
        t_2)
       (if (<= t_3 5e+61)
         (/
          -1.0
          (/
           t_0
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_0))))))
         (if (<= t_3 INFINITY)
           (/
            (*
             (sqrt (* 2.0 C))
             (sqrt (* 2.0 (* F (fma -4.0 (* A C) (* B_m B_m))))))
            t_2)
           (/ (- (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 t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) * (sqrt(t_0) * sqrt((2.0 * F)))) / t_2;
	} else if (t_3 <= -2e-195) {
		tmp = (sqrt((2.0 * fma(B_m, B_m, (-4.0 * (A * C))))) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) * sqrt(F))) / t_2;
	} else if (t_3 <= 5e+61) {
		tmp = -1.0 / (t_0 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) * sqrt((2.0 * (F * fma(-4.0, (A * C), (B_m * B_m)))))) / t_2;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) * Float64(sqrt(t_0) * sqrt(Float64(2.0 * F)))) / t_2);
	elseif (t_3 <= -2e-195)
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * Float64(sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) * sqrt(F))) / t_2);
	elseif (t_3 <= 5e+61)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_0))))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * sqrt(Float64(2.0 * Float64(F * fma(-4.0, Float64(A * C), Float64(B_m * B_m)))))) / t_2);
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-195], N[(N[(N[Sqrt[N[(2.0 * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 5e+61], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 9.1%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 21.0

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

   7. Simplified 21.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      13. pow1/2 N/A

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

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

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

      15. *-lowering-*.f64 27.0

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

   9. Applied egg-rr 27.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

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

   4. Applied egg-rr 55.6%

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

   6. Applied egg-rr 93.8%

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

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

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

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

   8. Applied egg-rr 97.6%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000018e61

   1. Initial program 16.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.9

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

   7. Simplified 18.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 26.3%

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

   if 5.00000000000000018e61 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 30.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 36.3

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

   7. Simplified 36.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{2 \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_4 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_5 := 2 \cdot \left(F \cdot t\_4\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{t\_3} \cdot \sqrt{2 \cdot F}\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_5}}{-t\_4}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{t\_3}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_3\right)\right)}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{t\_5}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_1))
        (t_3 (fma A (* C -4.0) (* B_m B_m)))
        (t_4 (fma -4.0 (* A C) (* B_m B_m)))
        (t_5 (* 2.0 (* F t_4))))
   (if (<= t_2 (- INFINITY))
     (/
      (*
       (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)))
       (* (sqrt t_3) (sqrt (* 2.0 F))))
      t_1)
     (if (<= t_2 -2e-195)
       (/
        (sqrt (* (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) t_5))
        (- t_4))
       (if (<= t_2 5e+61)
         (/
          -1.0
          (/
           t_3
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_3))))))
         (if (<= t_2 INFINITY)
           (/ (* (sqrt (* 2.0 C)) (sqrt t_5)) t_1)
           (/ (- (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 t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double t_3 = fma(A, (C * -4.0), (B_m * B_m));
	double t_4 = fma(-4.0, (A * C), (B_m * B_m));
	double t_5 = 2.0 * (F * t_4);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) * (sqrt(t_3) * sqrt((2.0 * F)))) / t_1;
	} else if (t_2 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_5)) / -t_4;
	} else if (t_2 <= 5e+61) {
		tmp = -1.0 / (t_3 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_3)))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) * sqrt(t_5)) / t_1;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_1)
	t_3 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_4 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_5 = Float64(2.0 * Float64(F * t_4))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) * Float64(sqrt(t_3) * sqrt(Float64(2.0 * F)))) / t_1);
	elseif (t_2 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_5)) / Float64(-t_4));
	elseif (t_2 <= 5e+61)
		tmp = Float64(-1.0 / Float64(t_3 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_3))))));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * sqrt(t_5)) / t_1);
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(F * t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$3], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision], If[LessEqual[t$95$2, 5e+61], N[(-1.0 / N[(t$95$3 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 9.1%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 21.0

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

   7. Simplified 21.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. unpow-prod-down N/A

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

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

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

      8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

      13. pow1/2 N/A

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

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

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

      15. *-lowering-*.f64 27.0

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

   9. Applied egg-rr 27.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000018e61

   1. Initial program 16.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.9

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

   7. Simplified 18.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 26.3%

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

   if 5.00000000000000018e61 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 30.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 36.3

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

   7. Simplified 36.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \left(\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{2 \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_5 := 2 \cdot \left(F \cdot t\_4\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{C \cdot F} \cdot \sqrt{2}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_5}}{-t\_4}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_0\right)\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{t\_5}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (fma -4.0 (* A C) (* B_m B_m)))
        (t_5 (* 2.0 (* F t_4))))
   (if (<= t_3 (- INFINITY))
     (/
      (*
       (sqrt (* 2.0 (fma B_m B_m (* -4.0 (* A C)))))
       (* (sqrt (* C F)) (sqrt 2.0)))
      t_2)
     (if (<= t_3 -2e-195)
       (/
        (sqrt (* (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) t_5))
        (- t_4))
       (if (<= t_3 5e+61)
         (/
          -1.0
          (/
           t_0
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_0))))))
         (if (<= t_3 INFINITY)
           (/ (* (sqrt (* 2.0 C)) (sqrt t_5)) t_2)
           (/ (- (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 t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = fma(-4.0, (A * C), (B_m * B_m));
	double t_5 = 2.0 * (F * t_4);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * fma(B_m, B_m, (-4.0 * (A * C))))) * (sqrt((C * F)) * sqrt(2.0))) / t_2;
	} else if (t_3 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_5)) / -t_4;
	} else if (t_3 <= 5e+61) {
		tmp = -1.0 / (t_0 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_0)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) * sqrt(t_5)) / t_2;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_5 = Float64(2.0 * Float64(F * t_4))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * Float64(sqrt(Float64(C * F)) * sqrt(2.0))) / t_2);
	elseif (t_3 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_5)) / Float64(-t_4));
	elseif (t_3 <= 5e+61)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_0))))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * sqrt(t_5)) / t_2);
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(F * t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / (-t$95$4)), $MachinePrecision], If[LessEqual[t$95$3, 5e+61], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

   4. Applied egg-rr 1.0%

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

   6. Applied egg-rr 20.8%

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

   7. Taylor expanded in A around -inf

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

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

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 25.7

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

   9. Simplified 25.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000018e61

   1. Initial program 16.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.9

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

   7. Simplified 18.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 26.3%

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

   if 5.00000000000000018e61 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 30.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 36.3

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

   7. Simplified 36.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(\sqrt{C \cdot F} \cdot \sqrt{2}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := 2 \cdot \left(F \cdot t\_0\right)\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ t_6 := \frac{\sqrt{2 \cdot C} \cdot \sqrt{t\_1}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_1}}{-t\_0}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{t\_2}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_2\right)\right)}}}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* 2.0 (* F t_0)))
        (t_2 (fma A (* C -4.0) (* B_m B_m)))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4))
        (t_6 (/ (* (sqrt (* 2.0 C)) (sqrt t_1)) t_4)))
   (if (<= t_5 (- INFINITY))
     t_6
     (if (<= t_5 -2e-195)
       (/
        (sqrt (* (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) t_1))
        (- t_0))
       (if (<= t_5 5e+61)
         (/
          -1.0
          (/
           t_2
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_2))))))
         (if (<= t_5 INFINITY) t_6 (/ (- (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 t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = 2.0 * (F * t_0);
	double t_2 = fma(A, (C * -4.0), (B_m * B_m));
	double t_3 = (4.0 * A) * C;
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double t_6 = (sqrt((2.0 * C)) * sqrt(t_1)) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_5 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_1)) / -t_0;
	} else if (t_5 <= 5e+61) {
		tmp = -1.0 / (t_2 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_2)))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_6;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(2.0 * Float64(F * t_0))
	t_2 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	t_6 = Float64(Float64(sqrt(Float64(2.0 * C)) * sqrt(t_1)) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_6;
	elseif (t_5 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_1)) / Float64(-t_0));
	elseif (t_5 <= 5e+61)
		tmp = Float64(-1.0 / Float64(t_2 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_2))))));
	elseif (t_5 <= Inf)
		tmp = t_6;
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], t$95$6, If[LessEqual[t$95$5, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$5, 5e+61], N[(-1.0 / N[(t$95$2 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$6, N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or 5.00000000000000018e61 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 10.4%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 20.4%

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

   5. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.1

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

   7. Simplified 24.1%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 5.00000000000000018e61

   1. Initial program 16.7%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 14.3%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.9

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

   7. Simplified 18.9%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 26.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_2 := F \cdot \left(2 \cdot t\_1\right)\\ t_3 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_4 := \sqrt{t\_0} \cdot \frac{\sqrt{t\_2}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ t_5 := \left(4 \cdot A\right) \cdot C\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_5\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_5 - {B\_m}^{2}}\\ \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_6 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot \left(2 \cdot \left(F \cdot t\_3\right)\right)}}{-t\_3}\\ \mathbf{elif}\;t\_6 \leq 10^{-78}:\\ \;\;\;\;\frac{-1}{\frac{t\_1}{\sqrt{t\_0 \cdot t\_2}}}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)))
        (t_1 (fma A (* C -4.0) (* B_m B_m)))
        (t_2 (* F (* 2.0 t_1)))
        (t_3 (fma -4.0 (* A C) (* B_m B_m)))
        (t_4 (* (sqrt t_0) (/ (sqrt t_2) (fma B_m (- B_m) (* A (* 4.0 C))))))
        (t_5 (* (* 4.0 A) C))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_5) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_5 (pow B_m 2.0)))))
   (if (<= t_6 (- INFINITY))
     t_4
     (if (<= t_6 -2e-195)
       (/
        (sqrt
         (*
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))
          (* 2.0 (* F t_3))))
        (- t_3))
       (if (<= t_6 1e-78)
         (/ -1.0 (/ t_1 (sqrt (* t_0 t_2))))
         (if (<= t_6 INFINITY) t_4 (/ (- (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 t_0 = fma(B_m, (-0.5 * (B_m / A)), (2.0 * C));
	double t_1 = fma(A, (C * -4.0), (B_m * B_m));
	double t_2 = F * (2.0 * t_1);
	double t_3 = fma(-4.0, (A * C), (B_m * B_m));
	double t_4 = sqrt(t_0) * (sqrt(t_2) / fma(B_m, -B_m, (A * (4.0 * C))));
	double t_5 = (4.0 * A) * C;
	double t_6 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_5) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_5 - pow(B_m, 2.0));
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_6 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * (2.0 * (F * t_3)))) / -t_3;
	} else if (t_6 <= 1e-78) {
		tmp = -1.0 / (t_1 / sqrt((t_0 * t_2)));
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C))
	t_1 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_2 = Float64(F * Float64(2.0 * t_1))
	t_3 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_4 = Float64(sqrt(t_0) * Float64(sqrt(t_2) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C)))))
	t_5 = Float64(Float64(4.0 * A) * C)
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_5) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_5 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_6 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * Float64(2.0 * Float64(F * t_3)))) / Float64(-t_3));
	elseif (t_6 <= 1e-78)
		tmp = Float64(-1.0 / Float64(t_1 / sqrt(Float64(t_0 * t_2))));
	elseif (t_6 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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 * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$5), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$5 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, (-Infinity)], t$95$4, If[LessEqual[t$95$6, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$6, 1e-78], N[(-1.0 / N[(t$95$1 / N[Sqrt[N[(t$95$0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, Infinity], t$95$4, N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or 9.99999999999999999e-79 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 13.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 22.7%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 25.9

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

   7. Simplified 25.9%

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

   9. Applied egg-rr 24.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 9.99999999999999999e-79

   1. Initial program 12.2%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 9.7%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 17.1

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

   7. Simplified 17.1%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 25.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right)} \cdot \frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{-78}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right)} \cdot \frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := 2 \cdot \left(F \cdot t\_0\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\right) \cdot \sqrt{\frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_1}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq 10^{+96}:\\ \;\;\;\;\frac{-1}{\frac{t\_4}{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot t\_4\right)\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)} \cdot \sqrt{t\_1}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* 2.0 (* F t_0)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_4 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= t_3 (- INFINITY))
     (* (* (sqrt F) (- (/ (sqrt 2.0) B_m))) (sqrt (/ (* B_m (* B_m -0.5)) A)))
     (if (<= t_3 -2e-195)
       (/
        (sqrt (* (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) t_1))
        (- t_0))
       (if (<= t_3 1e+96)
         (/
          -1.0
          (/
           t_4
           (sqrt
            (* (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C)) (* F (* 2.0 t_4))))))
         (if (<= t_3 INFINITY)
           (/
            (* (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))) (sqrt t_1))
            (* -4.0 (* A (- C))))
           (/ (- (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 t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = 2.0 * (F * t_0);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_4 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(F) * -(sqrt(2.0) / B_m)) * sqrt(((B_m * (B_m * -0.5)) / A));
	} else if (t_3 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_1)) / -t_0;
	} else if (t_3 <= 1e+96) {
		tmp = -1.0 / (t_4 / sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * t_4)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) * sqrt(t_1)) / (-4.0 * (A * -C));
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(2.0 * Float64(F * t_0))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(F) * Float64(-Float64(sqrt(2.0) / B_m))) * sqrt(Float64(Float64(B_m * Float64(B_m * -0.5)) / A)));
	elseif (t_3 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_1)) / Float64(-t_0));
	elseif (t_3 <= 1e+96)
		tmp = Float64(-1.0 / Float64(t_4 / sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * t_4))))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) * sqrt(t_1)) / Float64(-4.0 * Float64(A * Float64(-C))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[F], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision] * N[Sqrt[N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, 1e+96], N[(-1.0 / N[(t$95$4 / N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * (-C)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in 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 10.3

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

   5. Simplified 10.3%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

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

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. /-lowering-/.f64 N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-lowering-*.f64 11.2

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

   10. Applied egg-rr 11.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.00000000000000005e96

   1. Initial program 18.9%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 16.5%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.6

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

   7. Simplified 18.6%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

   9. Applied egg-rr 25.8%

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

   if 1.00000000000000005e96 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 26.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 48.6%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 38.1

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

   7. Simplified 38.1%

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

   8. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 38.1

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

   10. Simplified 38.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{\frac{B \cdot \left(B \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+96}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := 2 \cdot \left(F \cdot t\_0\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\right) \cdot \sqrt{\frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot t\_1}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\right)\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right)} \cdot \sqrt{t\_1}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* 2.0 (* F t_0)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (* (* (sqrt F) (- (/ (sqrt 2.0) B_m))) (sqrt (/ (* B_m (* B_m -0.5)) A)))
     (if (<= t_3 -2e-195)
       (/
        (sqrt (* (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) t_1))
        (- t_0))
       (if (<= t_3 1e+96)
         (/
          (sqrt
           (*
            (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C))
            (* F (* 2.0 (fma A (* C -4.0) (* B_m B_m))))))
          (fma B_m (- B_m) (* A (* 4.0 C))))
         (if (<= t_3 INFINITY)
           (/
            (* (sqrt (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C))) (sqrt t_1))
            (* -4.0 (* A (- C))))
           (/ (- (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 t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = 2.0 * (F * t_0);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(F) * -(sqrt(2.0) / B_m)) * sqrt(((B_m * (B_m * -0.5)) / A));
	} else if (t_3 <= -2e-195) {
		tmp = sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * t_1)) / -t_0;
	} else if (t_3 <= 1e+96) {
		tmp = sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * fma(A, (C * -4.0), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(fma(-0.5, ((B_m * B_m) / A), (2.0 * C))) * sqrt(t_1)) / (-4.0 * (A * -C));
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(2.0 * Float64(F * t_0))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(F) * Float64(-Float64(sqrt(2.0) / B_m))) * sqrt(Float64(Float64(B_m * Float64(B_m * -0.5)) / A)));
	elseif (t_3 <= -2e-195)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * t_1)) / Float64(-t_0));
	elseif (t_3 <= 1e+96)
		tmp = Float64(sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C))) * sqrt(t_1)) / Float64(-4.0 * Float64(A * Float64(-C))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[F], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision] * N[Sqrt[N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-195], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, 1e+96], N[(N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * (-C)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in 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 10.3

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

   5. Simplified 10.3%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

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

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. /-lowering-/.f64 N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-lowering-*.f64 11.2

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

   10. Applied egg-rr 11.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2.0000000000000002e-195

   1. Initial program 96.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 96.7%

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

   if -2.0000000000000002e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 1.00000000000000005e96

   1. Initial program 18.9%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 16.5%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 18.6

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

   7. Simplified 18.6%

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

   9. Applied egg-rr 25.7%

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

   if 1.00000000000000005e96 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 26.3%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

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

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

   4. Applied egg-rr 48.6%

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

   5. Taylor expanded in A around -inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. *-lowering-*.f64 38.1

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

   7. Simplified 38.1%

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

   8. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 38.1

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

   10. Simplified 38.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 13.6

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

   5. Simplified 13.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 21.2

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

   7. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 21.2

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

   9. Applied egg-rr 21.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 21.2

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

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{\frac{B \cdot \left(B \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-195}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{-4 \cdot \left(A \cdot \left(-C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\right) \cdot \sqrt{\frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0)))))
   (if (<= t_1 -5e+164)
     (* (* (sqrt F) (- (/ (sqrt 2.0) B_m))) (sqrt (/ (* B_m (* B_m -0.5)) A)))
     (if (<= t_1 -4e-138)
       (*
        (sqrt
         (/
          (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
          (fma B_m B_m (* -4.0 (* A C)))))
        (- (sqrt 2.0)))
       (/ (- (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 t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (t_1 <= -5e+164) {
		tmp = (sqrt(F) * -(sqrt(2.0) / B_m)) * sqrt(((B_m * (B_m * -0.5)) / A));
	} else if (t_1 <= -4e-138) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma(B_m, B_m, (-4.0 * (A * C))))) * -sqrt(2.0);
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -5e+164)
		tmp = Float64(Float64(sqrt(F) * Float64(-Float64(sqrt(2.0) / B_m))) * sqrt(Float64(Float64(B_m * Float64(B_m * -0.5)) / A)));
	elseif (t_1 <= -4e-138)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+164], N[(N[(N[Sqrt[F], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision] * N[Sqrt[N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-138], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.9999999999999995e164

   1. Initial program 13.0%

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

   3. Taylor expanded in 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 14.5

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

   5. Simplified 14.5%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

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

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. /-lowering-/.f64 N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-lowering-*.f64 10.1

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

   10. Applied egg-rr 10.1%

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

   if -4.9999999999999995e164 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.00000000000000027e-138

   1. Initial program 96.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

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

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

   5. Simplified 97.9%

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

   if -4.00000000000000027e-138 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 11.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 9.6

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

   5. Simplified 9.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. /-lowering-/.f64 N/A

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

      17. sqrt-lowering-sqrt.f64 14.7

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

   7. Applied egg-rr 14.7%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      9. /-lowering-/.f64 14.6

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

   9. Applied egg-rr 14.6%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 14.6

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

   11. Applied egg-rr 14.6%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+164}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{\frac{B \cdot \left(B \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\right) \cdot \sqrt{\frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0)))))
   (if (<= t_1 (- INFINITY))
     (* (* (sqrt F) (- (/ (sqrt 2.0) B_m))) (sqrt (/ (* B_m (* B_m -0.5)) A)))
     (if (<= t_1 -4e-178)
       (/
        (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m)))))))
        (- 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 t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (sqrt(F) * -(sqrt(2.0) / B_m)) * sqrt(((B_m * (B_m * -0.5)) / A));
	} else if (t_1 <= -4e-178) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -sqrt(F) / 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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(F) * Float64(-Float64(sqrt(2.0) / B_m))) * sqrt(Float64(Float64(B_m * Float64(B_m * -0.5)) / A)));
	elseif (t_1 <= -4e-178)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Sqrt[F], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision] * N[Sqrt[N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-178], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.1%

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

   3. Taylor expanded in 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 10.3

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

   5. Simplified 10.3%

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

   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. *-lowering-*.f64 N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

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

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. associate-*r/ N/A

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

      14. /-lowering-/.f64 N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-lowering-*.f64 11.2

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

   10. Applied egg-rr 11.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.9999999999999998e-178

   1. Initial program 96.6%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 29.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if -3.9999999999999998e-178 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 8.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 9.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 9.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 15.2

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 15.2%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 15.2

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 15.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 15.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 15.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right) \cdot \sqrt{\frac{B \cdot \left(B \cdot -0.5\right)}{A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B_m 2.0) t_0) F))
           (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B_m 2.0)))
        -4e+124)
     (* (- (/ (sqrt 2.0) B_m)) (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))))
     (/ (- (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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0))) <= -4e+124) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * ((a + c) + sqrt(((b_m ** 2.0d0) + ((a - c) ** 2.0d0)))))) / (t_0 - (b_m ** 2.0d0))) <= (-4d+124)) then
        tmp = -(sqrt(2.0d0) / b_m) * sqrt((f * ((-0.5d0) * ((b_m * b_m) / a))))
    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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_0 - Math.pow(B_m, 2.0))) <= -4e+124) {
		tmp = -(Math.sqrt(2.0) / B_m) * Math.sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	} 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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if (math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_0 - math.pow(B_m, 2.0))) <= -4e+124:
		tmp = -(math.sqrt(2.0) / B_m) * math.sqrt((F * (-0.5 * ((B_m * B_m) / A))))
	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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))));
	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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * (-0.5 * ((B_m * B_m) / A))));
	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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+124], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999979e124

   1. Initial program 16.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 14.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 14.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 9.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 9.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]

   if -3.99999999999999979e124 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 25.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 12.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 12.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 16.4

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 16.4%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 16.3

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 16.3%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 16.3

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 16.3%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \frac{F \cdot \left(B\_m \cdot \left(B\_m \cdot -0.5\right)\right)}{A}} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B_m 2.0) t_0) F))
           (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B_m 2.0)))
        -4e+124)
     (* (sqrt (* 2.0 (/ (* F (* B_m (* B_m -0.5))) A))) (/ -1.0 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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0))) <= -4e+124) {
		tmp = sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) * (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * ((a + c) + sqrt(((b_m ** 2.0d0) + ((a - c) ** 2.0d0)))))) / (t_0 - (b_m ** 2.0d0))) <= (-4d+124)) then
        tmp = sqrt((2.0d0 * ((f * (b_m * (b_m * (-0.5d0)))) / a))) * ((-1.0d0) / 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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_0 - Math.pow(B_m, 2.0))) <= -4e+124) {
		tmp = Math.sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) * (-1.0 / 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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if (math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_0 - math.pow(B_m, 2.0))) <= -4e+124:
		tmp = math.sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) * (-1.0 / 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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * Float64(B_m * Float64(B_m * -0.5))) / A))) * Float64(-1.0 / 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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) * (-1.0 / 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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+124], N[(N[Sqrt[N[(2.0 * N[(N[(F * N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / 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 (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999979e124

   1. Initial program 16.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 14.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 14.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 9.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 9.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}}{B}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}}{\mathsf{neg}\left(B\right)}} \]

      3. div-inv N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(B\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}\right) \cdot \frac{1}{\mathsf{neg}\left(B\right)}} \]

      5. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \sqrt{2 \cdot \left(F \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \left(B \cdot B\right)}{A}}\right)} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      9. associate-*r/ N/A

         \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}{A}}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \sqrt{2 \cdot \color{blue}{\frac{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}{A}}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \frac{\color{blue}{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}}{A}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      12. associate-*r* N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}}{A}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}}{A}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot B\right)} \cdot B\right)}{A}} \cdot \frac{1}{\mathsf{neg}\left(B\right)} \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}{A}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(B\right)\right)\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}{A}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(B\right)\right)\right)} \]

      17. remove-double-neg N/A

          \[\leadsto \sqrt{2 \cdot \frac{F \cdot \left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}{A}} \cdot \frac{-1}{\color{blue}{B}} \]

   10. Applied egg-rr 9.9%

       \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{F \cdot \left(\left(-0.5 \cdot B\right) \cdot B\right)}{A}} \cdot \frac{-1}{B}} \]

   if -3.99999999999999979e124 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 25.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 12.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 12.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 16.4

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 16.4%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 16.3

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 16.3%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 16.3

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 16.3%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2 \cdot \frac{F \cdot \left(B \cdot \left(B \cdot -0.5\right)\right)}{A}} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{F \cdot \left(B\_m \cdot \left(B\_m \cdot -0.5\right)\right)}{A}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<=
        (/
         (sqrt
          (*
           (* 2.0 (* (- (pow B_m 2.0) t_0) F))
           (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
         (- t_0 (pow B_m 2.0)))
        -4e+124)
     (/ (sqrt (* 2.0 (/ (* F (* B_m (* B_m -0.5))) A))) (- 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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0))) <= -4e+124) {
		tmp = sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) / -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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * ((a + c) + sqrt(((b_m ** 2.0d0) + ((a - c) ** 2.0d0)))))) / (t_0 - (b_m ** 2.0d0))) <= (-4d+124)) then
        tmp = sqrt((2.0d0 * ((f * (b_m * (b_m * (-0.5d0)))) / a))) / -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 t_0 = (4.0 * A) * C;
	double tmp;
	if ((Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_0 - Math.pow(B_m, 2.0))) <= -4e+124) {
		tmp = Math.sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) / -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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if (math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_0 - math.pow(B_m, 2.0))) <= -4e+124:
		tmp = math.sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) / -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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * Float64(B_m * Float64(B_m * -0.5))) / A))) / Float64(-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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_0 - (B_m ^ 2.0))) <= -4e+124)
		tmp = sqrt((2.0 * ((F * (B_m * (B_m * -0.5))) / A))) / -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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e+124], N[(N[Sqrt[N[(2.0 * N[(N[(F * N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $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 (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999979e124

   1. Initial program 16.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 14.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 14.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 9.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 9.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}}{B}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}}}{\mathsf{neg}\left(B\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(F \cdot \left(\frac{-1}{2} \cdot \frac{B \cdot B}{A}\right)\right)}}}{\mathsf{neg}\left(B\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \color{blue}{\frac{\frac{-1}{2} \cdot \left(B \cdot B\right)}{A}}\right)}}{\mathsf{neg}\left(B\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{2 \cdot \color{blue}{\frac{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}{A}}}}{\mathsf{neg}\left(B\right)} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot \color{blue}{\frac{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}{A}}}}{\mathsf{neg}\left(B\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \frac{\color{blue}{F \cdot \left(\frac{-1}{2} \cdot \left(B \cdot B\right)\right)}}{A}}}{\mathsf{neg}\left(B\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\sqrt{2 \cdot \frac{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}}{A}}}{\mathsf{neg}\left(B\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \frac{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot B\right)}}{A}}}{\mathsf{neg}\left(B\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot \frac{F \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot B\right)} \cdot B\right)}{A}}}{\mathsf{neg}\left(B\right)} \]

      14. neg-lowering-neg.f64 9.9

          \[\leadsto \frac{\sqrt{2 \cdot \frac{F \cdot \left(\left(-0.5 \cdot B\right) \cdot B\right)}{A}}}{\color{blue}{-B}} \]

   10. Applied egg-rr 9.9%

       \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \frac{F \cdot \left(\left(-0.5 \cdot B\right) \cdot B\right)}{A}}}{-B}} \]

   if -3.99999999999999979e124 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 25.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 12.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 12.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 16.4

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 16.4%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 16.3

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 16.3%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 16.3

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 16.3%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{F \cdot \left(B \cdot \left(B \cdot -0.5\right)\right)}{A}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B\_m, -0.5 \cdot \frac{B\_m}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\right)\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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) 5e-99)
   (/
    (sqrt
     (*
      (fma B_m (* -0.5 (/ B_m A)) (* 2.0 C))
      (* F (* 2.0 (fma A (* C -4.0) (* B_m B_m))))))
    (fma B_m (- B_m) (* A (* 4.0 C))))
   (if (<= (pow B_m 2.0) 5e+262)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- 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) <= 5e-99) {
		tmp = sqrt((fma(B_m, (-0.5 * (B_m / A)), (2.0 * C)) * (F * (2.0 * fma(A, (C * -4.0), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (pow(B_m, 2.0) <= 5e+262) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -sqrt(F) / 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) <= 5e-99)
		tmp = Float64(sqrt(Float64(fma(B_m, Float64(-0.5 * Float64(B_m / A)), Float64(2.0 * C)) * Float64(F * Float64(2.0 * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif ((B_m ^ 2.0) <= 5e+262)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-99], N[(N[Sqrt[N[(N[(B$95$m * N[(-0.5 * N[(B$95$m / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+262], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999969e-99

   1. Initial program 22.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}^{\frac{1}{2}} \cdot {\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 24.4%

      \[\leadsto \frac{-\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{{B}^{2}}{A}, 2 \cdot C\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{{B}^{2}}{A}}, 2 \cdot C\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{B \cdot B}}{A}, 2 \cdot C\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 20.5

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, \color{blue}{2 \cdot C}\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Simplified 20.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied egg-rr 24.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(B, \frac{B}{A} \cdot -0.5, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(C \cdot 4\right)\right)}} \]

   if 4.99999999999999969e-99 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000008e262

   1. Initial program 45.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 17.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if 5.00000000000000008e262 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.5

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 20.5%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 31.7

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 31.7%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 31.6

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 31.6%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 31.6

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 31.6%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, -0.5 \cdot \frac{B}{A}, 2 \cdot C\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \frac{B\_m \cdot -0.5}{\frac{A}{B\_m}}}\\ \mathbf{elif}\;B\_m \leq 4.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (<= B_m 6.5e-50)
   (- (* (/ (sqrt 2.0) B_m) (sqrt (* F (/ (* B_m -0.5) (/ A B_m))))))
   (if (<= B_m 4.3e+131)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- 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 (B_m <= 6.5e-50) {
		tmp = -((sqrt(2.0) / B_m) * sqrt((F * ((B_m * -0.5) / (A / B_m)))));
	} else if (B_m <= 4.3e+131) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -sqrt(F) / 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 <= 6.5e-50)
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(Float64(B_m * -0.5) / Float64(A / B_m))))));
	elseif (B_m <= 4.3e+131)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.5e-50], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(B$95$m * -0.5), $MachinePrecision] / N[(A / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), If[LessEqual[B$95$m, 4.3e+131], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.49999999999999987e-50

   1. Initial program 23.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 4.3

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 4.3%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 6.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(B \cdot \frac{B}{A}\right)}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot \frac{B}{A}\right)}}\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\left(\frac{-1}{2} \cdot B\right) \cdot \color{blue}{\frac{1}{\frac{A}{B}}}\right)}\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{\frac{-1}{2} \cdot B}{\frac{A}{B}}}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{\frac{-1}{2} \cdot B}{\frac{A}{B}}}}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{\color{blue}{\frac{-1}{2} \cdot B}}{\frac{A}{B}}}\right) \]

      7. /-lowering-/.f64 7.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{-0.5 \cdot B}{\color{blue}{\frac{A}{B}}}} \]

   10. Applied egg-rr 7.9%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5 \cdot B}{\frac{A}{B}}}} \]

   if 6.49999999999999987e-50 < B < 4.3000000000000001e131

   1. Initial program 45.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if 4.3000000000000001e131 < B

   1. Initial program 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 48.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 78.1

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 78.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 77.9

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 77.9%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 77.9

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 77.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \frac{B \cdot -0.5}{\frac{A}{B}}}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.4 \cdot 10^{-50}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(\frac{B\_m}{A} \cdot \left(B\_m \cdot -0.5\right)\right)}\\ \mathbf{elif}\;B\_m \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (<= B_m 8.4e-50)
   (* (- (/ (sqrt 2.0) B_m)) (sqrt (* F (* (/ B_m A) (* B_m -0.5)))))
   (if (<= B_m 1.9e+131)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- 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 (B_m <= 8.4e-50) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * ((B_m / A) * (B_m * -0.5))));
	} else if (B_m <= 1.9e+131) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -sqrt(F) / 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 <= 8.4e-50)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(Float64(B_m / A) * Float64(B_m * -0.5)))));
	elseif (B_m <= 1.9e+131)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.4e-50], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(B$95$m / A), $MachinePrecision] * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.9e+131], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.4000000000000003e-50

   1. Initial program 23.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 4.3

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 4.3%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 6.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(B \cdot \frac{B}{A}\right)}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot \frac{B}{A}\right)}}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot \frac{B}{A}\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot B\right)} \cdot \frac{B}{A}\right)}\right) \]

      5. /-lowering-/.f64 7.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\left(-0.5 \cdot B\right) \cdot \color{blue}{\frac{B}{A}}\right)} \]

   10. Applied egg-rr 7.9%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(-0.5 \cdot B\right) \cdot \frac{B}{A}\right)}} \]

   if 8.4000000000000003e-50 < B < 1.9000000000000002e131

   1. Initial program 45.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if 1.9000000000000002e131 < B

   1. Initial program 3.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 48.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 78.1

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 78.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 77.9

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 77.9%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 77.9

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 77.9%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.4 \cdot 10^{-50}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(\frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9 \cdot 10^{-50}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B\_m}\right) \cdot \sqrt{F \cdot \left(\frac{B\_m}{A} \cdot \left(B\_m \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 (<= B_m 9e-50)
   (* (- (/ (sqrt 2.0) B_m)) (sqrt (* F (* (/ B_m A) (* B_m -0.5)))))
   (/ (- (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 (B_m <= 9e-50) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * ((B_m / A) * (B_m * -0.5))));
	} 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 <= 9d-50) then
        tmp = -(sqrt(2.0d0) / b_m) * sqrt((f * ((b_m / a) * (b_m * (-0.5d0)))))
    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 (B_m <= 9e-50) {
		tmp = -(Math.sqrt(2.0) / B_m) * Math.sqrt((F * ((B_m / A) * (B_m * -0.5))));
	} 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 B_m <= 9e-50:
		tmp = -(math.sqrt(2.0) / B_m) * math.sqrt((F * ((B_m / A) * (B_m * -0.5))))
	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 <= 9e-50)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(F * Float64(Float64(B_m / A) * Float64(B_m * -0.5)))));
	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 <= 9e-50)
		tmp = -(sqrt(2.0) / B_m) * sqrt((F * ((B_m / A) * (B_m * -0.5))));
	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[B$95$m, 9e-50], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(B$95$m / A), $MachinePrecision] * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 B < 8.99999999999999924e-50

   1. Initial program 23.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 4.3

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   5. Simplified 4.3%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}\right) \]

      4. *-lowering-*.f64 6.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(-0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \]

   8. Simplified 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A}\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(B \cdot \frac{B}{A}\right)}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot \frac{B}{A}\right)}}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot B\right) \cdot \frac{B}{A}\right)}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot B\right)} \cdot \frac{B}{A}\right)}\right) \]

      5. /-lowering-/.f64 7.9

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(\left(-0.5 \cdot B\right) \cdot \color{blue}{\frac{B}{A}}\right)} \]

   10. Applied egg-rr 7.9%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(\left(-0.5 \cdot B\right) \cdot \frac{B}{A}\right)}} \]

   if 8.99999999999999924e-50 < B

   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 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 39.5

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

      17. sqrt-lowering-sqrt.f64 54.3

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 54.3%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 54.2

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 54.2%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
   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 54.2

           \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 54.2%

       \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-50}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(\frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.3% 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 23.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 11.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. sqrt-div N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

   17. sqrt-lowering-sqrt.f64 15.5

       \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

7. Applied egg-rr 15.5%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   9. /-lowering-/.f64 15.5

      \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 15.5%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
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 15.5

        \[\leadsto \frac{-\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

11. Applied egg-rr 15.5%

    \[\leadsto \color{blue}{\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]
12. Add Preprocessing

Alternative 19: 35.3% 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 23.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 11.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. sqrt-div N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{B}}}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{1}{\sqrt{B}}}\right) \]

   17. sqrt-lowering-sqrt.f64 15.5

       \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

7. Applied egg-rr 15.5%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   9. /-lowering-/.f64 15.5

      \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 15.5%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

10. Final simplification 15.5%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 20: 26.7% 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 23.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 11.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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 11.5

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 11.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Add Preprocessing

Alternative 21: 26.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ F B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((2.0 * (F / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((2.0d0 * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(2.0 * Float64(F / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 23.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 11.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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 11.5

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 11.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F}{B} \cdot 2}}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F}{B} \cdot 2}}\right) \]

   4. /-lowering-/.f64 11.5

      \[\leadsto -\sqrt{\color{blue}{\frac{F}{B}} \cdot 2} \]

9. Applied egg-rr 11.5%

   \[\leadsto -\sqrt{\color{blue}{\frac{F}{B} \cdot 2}} \]

10. Final simplification 11.5%

    \[\leadsto -\sqrt{2 \cdot \frac{F}{B}} \]
11. Add Preprocessing

Alternative 22: 26.7% 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 23.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 11.5

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.5%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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 11.5

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 11.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
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 11.5

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 11.5%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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))))