ABCF->ab-angle a

Percentage Accurate: 19.5% → 62.5%

Time: 19.8s

Alternatives: 21

Speedup: 16.9×

• could not determine a ground truth

  1. A = -6.220805173741362e+65
  2. B = -5.738260805902138e-76
  3. C = 8.97812798168438e-216
  4. F = 4.350858839923357e+104

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 62.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\\ t_4 := t\_2 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{-1}{C - A}, \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(2 \cdot t\_1\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{t\_0} \cdot \left(\sqrt{F \cdot t\_0} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (* 2.0 (* (- (pow B_m 2.0) t_2) F)))
        (t_4 (- t_2 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt (* t_3 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (* (* (sqrt t_0) (- (* 2.0 (sqrt F)))) (/ (sqrt C) t_1))
     (if (<= t_5 -1e-171)
       (/
        (sqrt
         (*
          t_3
          (fma
           (* (+ A C) (- A C))
           (/ -1.0 (- C A))
           (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        t_4)
       (if (<= t_5 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* 2.0 t_1))))
          t_4)
         (if (<= t_5 INFINITY)
           (* (sqrt C) (* (/ 1.0 t_0) (* (sqrt (* F t_0)) -2.0)))
           (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = (4.0 * A) * C;
	double t_3 = 2.0 * ((pow(B_m, 2.0) - t_2) * F);
	double t_4 = t_2 - pow(B_m, 2.0);
	double t_5 = sqrt((t_3 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * -(2.0 * sqrt(F))) * (sqrt(C) / t_1);
	} else if (t_5 <= -1e-171) {
		tmp = sqrt((t_3 * fma(((A + C) * (A - C)), (-1.0 / (C - A)), sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_4;
	} else if (t_5 <= 0.0) {
		tmp = (sqrt(F) * sqrt((fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (2.0 * t_1)))) / t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt(C) * ((1.0 / t_0) * (sqrt((F * t_0)) * -2.0));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F))
	t_4 = Float64(t_2 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_1));
	elseif (t_5 <= -1e-171)
		tmp = Float64(sqrt(Float64(t_3 * fma(Float64(Float64(A + C) * Float64(A - C)), Float64(-1.0 / Float64(C - A)), sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / t_4);
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(2.0 * t_1)))) / t_4);
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(C) * Float64(Float64(1.0 / t_0) * Float64(sqrt(Float64(F * t_0)) * -2.0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 * 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]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -1e-171], N[(N[Sqrt[N[(t$95$3 * N[(N[(N[(A + C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(C - A), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[C], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.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))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 29.1

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

   5. Simplified 29.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

   7. Applied egg-rr 41.4%

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

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.6%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. /-rgt-identity N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 36.9%

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

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 43.1%

   \[\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:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{-1}{C - A}, \sqrt{\mathsf{fma}\left(A - C, A - 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 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(2 \cdot t\_1\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{t\_0} \cdot \left(\sqrt{F \cdot t\_0} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (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_3)))
   (if (<= t_4 (- INFINITY))
     (* (* (sqrt t_0) (- (* 2.0 (sqrt F)))) (/ (sqrt C) t_1))
     (if (<= t_4 -1e-171)
       (/
        (sqrt
         (*
          (* t_1 (* 2.0 F))
          (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        (- t_1))
       (if (<= t_4 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* 2.0 t_1))))
          t_3)
         (if (<= t_4 INFINITY)
           (* (sqrt C) (* (/ 1.0 t_0) (* (sqrt (* F t_0)) -2.0)))
           (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = 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_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * -(2.0 * sqrt(F))) * (sqrt(C) / t_1);
	} else if (t_4 <= -1e-171) {
		tmp = sqrt(((t_1 * (2.0 * F)) * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / -t_1;
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt((fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (2.0 * t_1)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(C) * ((1.0 / t_0) * (sqrt((F * t_0)) * -2.0));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = 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)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_1));
	elseif (t_4 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(t_1 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / Float64(-t_1));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(2.0 * t_1)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(C) * Float64(Float64(1.0 / t_0) * Float64(sqrt(Float64(F * t_0)) * -2.0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-171], N[(N[Sqrt[N[(N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * 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]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[C], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.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))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 29.1

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

   5. Simplified 29.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

   7. Applied egg-rr 41.4%

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

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.6%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. /-rgt-identity N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 36.9%

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

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 43.1%

   \[\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:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \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, A \cdot \left(C \cdot -4\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 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\sqrt{t\_0} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-t\_1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot t\_1\right) \cdot \left(2 \cdot C\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{t\_0} \cdot \left(\sqrt{F \cdot t\_0} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (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_3)))
   (if (<= t_4 (- INFINITY))
     (* (* (sqrt t_0) (- (* 2.0 (sqrt F)))) (/ (sqrt C) t_1))
     (if (<= t_4 -1e-171)
       (/
        (sqrt
         (*
          (* t_1 (* 2.0 F))
          (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        (- t_1))
       (if (<= t_4 0.0)
         (/ (* (sqrt F) (sqrt (* (* 2.0 t_1) (* 2.0 C)))) t_3)
         (if (<= t_4 INFINITY)
           (* (sqrt C) (* (/ 1.0 t_0) (* (sqrt (* F t_0)) -2.0)))
           (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = 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_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * -(2.0 * sqrt(F))) * (sqrt(C) / t_1);
	} else if (t_4 <= -1e-171) {
		tmp = sqrt(((t_1 * (2.0 * F)) * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / -t_1;
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((2.0 * t_1) * (2.0 * C)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(C) * ((1.0 / t_0) * (sqrt((F * t_0)) * -2.0));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = 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)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_1));
	elseif (t_4 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(t_1 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / Float64(-t_1));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * t_1) * Float64(2.0 * C)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(C) * Float64(Float64(1.0 / t_0) * Float64(sqrt(Float64(F * t_0)) * -2.0)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-171], N[(N[Sqrt[N[(N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * 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]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[C], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.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))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 25.5

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

   5. Simplified 25.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

   7. Applied egg-rr 34.6%

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

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 23.6%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 32.4

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

   5. Simplified 32.4%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.6%

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

      1. *-commutative N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. div-inv N/A

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

      5. /-rgt-identity N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 36.9%

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

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 42.2%

   \[\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:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\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 -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \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, A \cdot \left(C \cdot -4\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 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{C} \cdot \left(\frac{1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := t\_0 \cdot \left(2 \cdot F\right)\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1}}{t\_2} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* t_0 (* 2.0 F)))
        (t_2 (- t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (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_3 (pow B_m 2.0)))))
   (if (<= t_4 (- INFINITY))
     (*
      (* (sqrt (fma B_m B_m (* C (* A -4.0)))) (- (* 2.0 (sqrt F))))
      (/ (sqrt C) t_0))
     (if (<= t_4 -1e-171)
       (/
        (sqrt (* t_1 (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        t_2)
       (if (<= t_4 INFINITY)
         (* (/ (sqrt t_1) t_2) (sqrt (fma 2.0 C (/ (* B_m (* B_m -0.5)) A))))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = t_0 * (2.0 * F);
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = 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_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(B_m, B_m, (C * (A * -4.0)))) * -(2.0 * sqrt(F))) * (sqrt(C) / t_0);
	} else if (t_4 <= -1e-171) {
		tmp = sqrt((t_1 * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_1) / t_2) * sqrt(fma(2.0, C, ((B_m * (B_m * -0.5)) / A)));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(t_0 * Float64(2.0 * F))
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = 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)))))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_0));
	elseif (t_4 <= -1e-171)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_1) / t_2) * sqrt(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = 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] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-171], N[(N[Sqrt[N[(t$95$1 * 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]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Sqrt[N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.4%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.5

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

   5. Simplified 30.5%

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

   7. Applied egg-rr 32.9%

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

   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)))

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_0\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- 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 (fma B_m B_m (* C (* A -4.0)))) (- (* 2.0 (sqrt F))))
      (/ (sqrt C) t_0))
     (if (<= t_3 -1e-171)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        t_1)
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_0))))
          t_1)
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -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(fma(B_m, B_m, (C * (A * -4.0)))) * -(2.0 * sqrt(F))) * (sqrt(C) / t_0);
	} else if (t_3 <= -1e-171) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_0)))) / t_1;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-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(fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_0));
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / t_1);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_0)))) / t_1);
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, 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[N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * 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]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.4%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.5

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

   5. Simplified 30.5%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   7. Applied egg-rr 30.5%

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

   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)))

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_1\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (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 (fma B_m B_m (* C (* A -4.0)))) (- (* 2.0 (sqrt F))))
      (/ (sqrt C) t_1))
     (if (<= t_3 -1e-171)
       (/
        (sqrt
         (*
          (* 2.0 (* F t_0))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_1))))
          (- t_1))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	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(fma(B_m, B_m, (C * (A * -4.0)))) * -(2.0 * sqrt(F))) * (sqrt(C) / t_1);
	} else if (t_3 <= -1e-171) {
		tmp = sqrt(((2.0 * (F * t_0)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_1)))) / -t_1;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	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(fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_1));
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_1)))) / Float64(-t_1));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $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[N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * 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]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. /-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 99.3%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.4%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.5

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

   5. Simplified 30.5%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   7. Applied egg-rr 30.5%

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

   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)))

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_1\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (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 (fma B_m B_m (* C (* A -4.0)))) (- (* 2.0 (sqrt F))))
      (/ (sqrt C) t_1))
     (if (<= t_3 -1e-171)
       (*
        (sqrt (* (* 2.0 (* F t_0)) (+ (+ A C) (sqrt (fma B_m B_m (* C C))))))
        (/ -1.0 t_0))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_1))))
          (- t_1))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	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(fma(B_m, B_m, (C * (A * -4.0)))) * -(2.0 * sqrt(F))) * (sqrt(C) / t_1);
	} else if (t_3 <= -1e-171) {
		tmp = sqrt(((2.0 * (F * t_0)) * ((A + C) + sqrt(fma(B_m, B_m, (C * C)))))) * (-1.0 / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_1)))) / -t_1;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	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(fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_1));
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(-1.0 / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_1)))) / Float64(-t_1));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $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[N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 36.7%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.9

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

   9. Applied egg-rr 46.9%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in A around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 86.0

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

   7. Simplified 86.0%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.4%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.5

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

   5. Simplified 30.5%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   7. Applied egg-rr 30.5%

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

   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)))

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. pow1/2 N/A

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

      6. pow-prod-down N/A

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

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

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

      8. pow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow1/2 N/A

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

      11. sqrt-unprod N/A

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 26.3

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

   7. Applied egg-rr 26.3%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (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_1 (pow B_m 2.0)))))
   (if (<= t_2 -2e+154)
     (*
      (* (sqrt (fma B_m B_m (* C (* A -4.0)))) (- (* 2.0 (sqrt F))))
      (/ (sqrt C) t_0))
     (if (<= t_2 -5e-165)
       (*
        (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)))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_0))))
          (- t_0))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = 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_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -2e+154) {
		tmp = (sqrt(fma(B_m, B_m, (C * (A * -4.0)))) * -(2.0 * sqrt(F))) * (sqrt(C) / t_0);
	} else if (t_2 <= -5e-165) {
		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 if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_0)))) / -t_0;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = 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)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -2e+154)
		tmp = Float64(Float64(sqrt(fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(-Float64(2.0 * sqrt(F)))) * Float64(sqrt(C) / t_0));
	elseif (t_2 <= -5e-165)
		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)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = 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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+154], N[(N[(N[Sqrt[N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-165], 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], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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))) < -2.00000000000000007e154

   1. Initial program 11.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 26.3

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

   5. Simplified 26.3%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*r* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

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

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

   7. Applied egg-rr 37.3%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. *-commutative N/A

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

      7. unpow-prod-down N/A

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

      8. unpow1/2 N/A

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

      9. metadata-eval N/A

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

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

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

      11. pow1/2 N/A

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

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

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

      13. pow1/2 N/A

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

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

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

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

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-lowering-*.f64 46.7

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

   9. Applied egg-rr 46.7%

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

   if -2.00000000000000007e154 < (/.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))) < -4.99999999999999981e-165

   1. Initial program 99.4%

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

   3. Taylor expanded in 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 95.9%

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

   if -4.99999999999999981e-165 < (/.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 13.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.2

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

   5. Simplified 30.2%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   7. Applied egg-rr 30.2%

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

   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)))

   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 17.8

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

   5. Simplified 17.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 26.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-2 \cdot \sqrt{F}\right)\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot t\_0} \cdot -2\right) \cdot \frac{\sqrt{C}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B\_m \cdot \left(B\_m \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot t\_1\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (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 -2e+154)
     (* (* (sqrt (* F t_0)) -2.0) (/ (sqrt C) t_0))
     (if (<= t_3 -5e-165)
       (*
        (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)))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* 2.0 (* (fma 2.0 C (/ (* B_m (* B_m -0.5)) A)) (* F t_1))))
          (- t_1))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	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 <= -2e+154) {
		tmp = (sqrt((F * t_0)) * -2.0) * (sqrt(C) / t_0);
	} else if (t_3 <= -5e-165) {
		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 if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((2.0 * (fma(2.0, C, ((B_m * (B_m * -0.5)) / A)) * (F * t_1)))) / -t_1;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	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 <= -2e+154)
		tmp = Float64(Float64(sqrt(Float64(F * t_0)) * -2.0) * Float64(sqrt(C) / t_0));
	elseif (t_3 <= -5e-165)
		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)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(2.0 * Float64(fma(2.0, C, Float64(Float64(B_m * Float64(B_m * -0.5)) / A)) * Float64(F * t_1)))) / Float64(-t_1));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $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, -2e+154], N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-165], 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], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C + N[(N[(B$95$m * N[(B$95$m * -0.5), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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))) < -2.00000000000000007e154

   1. Initial program 11.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 26.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 26.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      3. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C}}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{C}}{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot -4\right) \cdot A}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F} \cdot \sqrt{4}}\right)\right) \]

   9. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{\frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot -2\right)} \]

   if -2.00000000000000007e154 < (/.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))) < -4.99999999999999981e-165

   1. Initial program 99.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 95.9%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   if -4.99999999999999981e-165 < (/.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 13.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\color{blue}{\frac{-1}{2} \cdot {B}^{2}}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\frac{-1}{2} \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. *-lowering-*.f64 30.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 30.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \frac{\frac{-1}{2} \cdot \left(B \cdot B\right)}{A}\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \frac{\frac{-1}{2} \cdot \left(B \cdot B\right)}{A}\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \frac{\frac{-1}{2} \cdot \left(B \cdot B\right)}{A}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied egg-rr 30.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   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)))

   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 17.8

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 26.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(2, C, \frac{B \cdot \left(B \cdot -0.5\right)}{A}\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot t\_0} \cdot -2\right) \cdot \frac{\sqrt{C}}{t\_0}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{t\_3 \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-t\_3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (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_1 (pow B_m 2.0))))
        (t_3 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= t_2 -2e+154)
     (* (* (sqrt (* F t_0)) -2.0) (/ (sqrt C) t_0))
     (if (<= t_2 -5e-134)
       (*
        (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)))
       (if (<= t_2 INFINITY)
         (* (sqrt 2.0) (/ (sqrt (* t_3 (* F (* 2.0 C)))) (- t_3)))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = 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_1 - pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (t_2 <= -2e+154) {
		tmp = (sqrt((F * t_0)) * -2.0) * (sqrt(C) / t_0);
	} else if (t_2 <= -5e-134) {
		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 if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(2.0) * (sqrt((t_3 * (F * (2.0 * C)))) / -t_3);
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = 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)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_2 <= -2e+154)
		tmp = Float64(Float64(sqrt(Float64(F * t_0)) * -2.0) * Float64(sqrt(C) / t_0));
	elseif (t_2 <= -5e-134)
		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)));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(t_3 * Float64(F * Float64(2.0 * C)))) / Float64(-t_3)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = 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] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+154], N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-134], 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], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(t$95$3 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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))) < -2.00000000000000007e154

   1. Initial program 11.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 26.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 26.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      3. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C}}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{C}}{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot -4\right) \cdot A}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F} \cdot \sqrt{4}}\right)\right) \]

   9. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{\frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot -2\right)} \]

   if -2.00000000000000007e154 < (/.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))) < -5.0000000000000003e-134

   1. Initial program 99.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 99.4%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   if -5.0000000000000003e-134 < (/.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 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 29.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. sqrt-prod N/A

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{\sqrt{2} \cdot {\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1}} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{2}}}{-1} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   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)))

   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 17.8

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 26.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot t\_1} \cdot -2\right) \cdot \frac{\sqrt{C}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (fma B_m B_m (* C (* A -4.0))))
        (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 -2e+154)
     (* (* (sqrt (* F t_1)) -2.0) (/ (sqrt C) t_1))
     (if (<= t_3 -5e-134)
       (*
        (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)))
       (if (<= t_3 INFINITY)
         (* (sqrt (* C (* 4.0 (* F t_0)))) (/ -1.0 t_0))
         (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(B_m, B_m, (C * (A * -4.0)));
	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 <= -2e+154) {
		tmp = (sqrt((F * t_1)) * -2.0) * (sqrt(C) / t_1);
	} else if (t_3 <= -5e-134) {
		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 if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) * (-1.0 / t_0);
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	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 <= -2e+154)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * -2.0) * Float64(sqrt(C) / t_1));
	elseif (t_3 <= -5e-134)
		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)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $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, -2e+154], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-134], 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], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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))) < -2.00000000000000007e154

   1. Initial program 11.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 26.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 26.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 37.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
   8. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\mathsf{neg}\left(-1\right)}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)}{\color{blue}{1}} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      3. /-rgt-identity N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \cdot \frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{C}}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C}}}{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{C}}{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot -4\right) \cdot A}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(-4 \cdot A\right)}\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}\right)\right) \]

      13. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot F} \cdot \sqrt{4}}\right)\right) \]

   9. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{\frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot -2\right)} \]

   if -2.00000000000000007e154 < (/.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))) < -5.0000000000000003e-134

   1. Initial program 99.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 99.4%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   if -5.0000000000000003e-134 < (/.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 15.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 29.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}} \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. frac-2neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \color{blue}{\frac{-1}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      8. cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\color{blue}{B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      10. +-commutative N/A

          \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{\color{blue}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)} \cdot \frac{-1}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

   7. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 17.8

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 26.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 26.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot -2\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)} \cdot \left(B\_m \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-32)
     (/ (sqrt (* C (* 4.0 (* F t_0)))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+262)
       (*
        (/ -1.0 (fma -4.0 (* A C) (* B_m B_m)))
        (* B_m (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C)))))))))
       (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+262) {
		tmp = (-1.0 / fma(-4.0, (A * C), (B_m * B_m))) * (B_m * (sqrt(2.0) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))))));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+262)
		tmp = Float64(Float64(-1.0 / fma(-4.0, Float64(A * C), Float64(B_m * B_m))) * Float64(B_m * Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+262], N[(N[(-1.0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 29.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64)) < 1e262

   1. Initial program 34.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. 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. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}} \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{\color{blue}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   4. Applied egg-rr 34.1%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{\left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{\left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \left(B \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(B \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{\sqrt{{B}^{2} + {C}^{2}}}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {C}^{2}\right)}}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      11. unpow2 N/A

          \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      12. *-lowering-*.f64 17.4

          \[\leadsto \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)}\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   7. Simplified 17.4%

      \[\leadsto \color{blue}{\left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 1e262 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 28.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 28.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 42.7

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 42.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-32)
     (/ (sqrt (* C (* 4.0 (* F t_0)))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+262)
       (/
        (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m)))))))
        (- B_m))
       (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+262) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+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(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+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[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 29.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64)) < 1e262

   1. Initial program 34.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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 17.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if 1e262 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 28.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 28.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 42.7

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 42.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)} \cdot \sqrt{\left(C \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot F\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (*
    (/ -1.0 (fma -4.0 (* A C) (* B_m B_m)))
    (sqrt (* (* C (* A -16.0)) (* C F))))
   (if (<= (pow B_m 2.0) 1e+262)
     (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) (- B_m))
     (- (/ (sqrt (* 2.0 F)) (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = (-1.0 / fma(-4.0, (A * C), (B_m * B_m))) * sqrt(((C * (A * -16.0)) * (C * F)));
	} else if (pow(B_m, 2.0) <= 1e+262) {
		tmp = (sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / -B_m;
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(Float64(-1.0 / fma(-4.0, Float64(A * C), Float64(B_m * B_m))) * sqrt(Float64(Float64(C * Float64(A * -16.0)) * Float64(C * F))));
	elseif ((B_m ^ 2.0) <= 1e+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(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[(-1.0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(C * N[(A * -16.0), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+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[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}} \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{\color{blue}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   4. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      5. *-lowering-*.f64 14.8

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   7. Simplified 14.8%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(C \cdot \left(C \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right)} \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-16 \cdot A\right)} \cdot C\right) \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      8. *-lowering-*.f64 28.6

         \[\leadsto \sqrt{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   9. Applied egg-rr 28.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64)) < 1e262

   1. Initial program 34.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   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 17.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if 1e262 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 28.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 28.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 42.7

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 42.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(C \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot F\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)} \cdot \sqrt{\left(C \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (*
    (/ -1.0 (fma -4.0 (* A C) (* B_m B_m)))
    (sqrt (* (* C (* A -16.0)) (* C F))))
   (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = (-1.0 / fma(-4.0, (A * C), (B_m * B_m))) * sqrt(((C * (A * -16.0)) * (C * F)));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(Float64(-1.0 / fma(-4.0, Float64(A * C), Float64(B_m * B_m))) * sqrt(Float64(Float64(C * Float64(A * -16.0)) * Float64(C * F))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[(-1.0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(C * N[(A * -16.0), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. 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. div-inv N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}} \cdot \frac{1}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. frac-2neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)} \]

      6. remove-double-neg N/A

         \[\leadsto \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{\color{blue}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \frac{-1}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   4. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right)\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      5. *-lowering-*.f64 14.8

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   7. Simplified 14.8%

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{\left(-16 \cdot A\right) \cdot \color{blue}{\left(C \cdot \left(C \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(C \cdot F\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right)} \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \sqrt{\left(\color{blue}{\left(-16 \cdot A\right)} \cdot C\right) \cdot \left(C \cdot F\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

      8. *-lowering-*.f64 28.6

         \[\leadsto \sqrt{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   9. Applied egg-rr 28.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(-16 \cdot A\right) \cdot C\right) \cdot \left(F \cdot C\right)}} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 22.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 22.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 30.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 30.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(C \cdot \left(A \cdot -16\right)\right) \cdot \left(C \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (* -2.0 (sqrt (/ (* C F) (fma B_m B_m (* -4.0 (* A C))))))
   (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = -2.0 * sqrt(((C * F) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 29.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{C \cdot F}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}} \]

      6. unpow2 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      7. metadata-eval N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}} \]

      10. *-lowering-*.f64 21.8

          \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}} \]

   8. Simplified 21.8%

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 22.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 22.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 30.3

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 30.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 8.8e+192)
   (- (/ (sqrt (* 2.0 F)) (sqrt B_m)))
   (* -2.0 (/ (sqrt (* C F)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8.8e+192) {
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 8.8d+192) then
        tmp = -(sqrt((2.0d0 * f)) / sqrt(b_m))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8.8e+192) {
		tmp = -(Math.sqrt((2.0 * F)) / Math.sqrt(B_m));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 8.8e+192:
		tmp = -(math.sqrt((2.0 * F)) / math.sqrt(B_m))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 8.8e+192)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * F)) / sqrt(B_m)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 8.8e+192)
		tmp = -(sqrt((2.0 * F)) / sqrt(B_m));
	else
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 8.8e+192], (-N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 8.8000000000000003e192

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 15.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 20.1

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 20.1%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]

   if 8.8000000000000003e192 < C

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 35.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 35.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{1 \cdot \sqrt{C \cdot F}}{B}} \]

      3. *-lft-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F}}{B}} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      6. *-commutative N/A

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

      7. *-lowering-*.f64 8.8

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   10. Simplified 8.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 8.8 \cdot 10^{+192}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot F}}{\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.25 \cdot 10^{+193}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.25e+193)
   (- (* (sqrt F) (sqrt (/ 2.0 B_m))))
   (* -2.0 (/ (sqrt (* C F)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.25e+193) {
		tmp = -(sqrt(F) * sqrt((2.0 / B_m)));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 1.25d+193) then
        tmp = -(sqrt(f) * sqrt((2.0d0 / b_m)))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.25e+193) {
		tmp = -(Math.sqrt(F) * Math.sqrt((2.0 / B_m)));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 1.25e+193:
		tmp = -(math.sqrt(F) * math.sqrt((2.0 / B_m)))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 1.25e+193)
		tmp = Float64(-Float64(sqrt(F) * sqrt(Float64(2.0 / B_m))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 1.25e+193)
		tmp = -(sqrt(F) * sqrt((2.0 / B_m)));
	else
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 1.25e+193], (-N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.24999999999999993e193

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 15.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{B}}} \cdot \sqrt{2}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}}}{\sqrt{B}} \cdot \sqrt{2}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{F}^{\frac{1}{2}} \cdot \sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{{2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      6. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(F \cdot 2\right)}^{\frac{1}{2}}}{\sqrt{B}}}\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}}}{\sqrt{B}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{{F}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}}{\sqrt{B}}\right) \]

      10. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      14. sqrt-lowering-sqrt.f64 20.1

          \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   7. Applied egg-rr 20.1%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F} \cdot \sqrt{2}}}{\sqrt{B}}\right) \]

      2. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{2}}{\sqrt{B}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      9. /-lowering-/.f64 20.0

         \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 20.0%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

   if 1.24999999999999993e193 < C

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 35.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 35.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{1 \cdot \sqrt{C \cdot F}}{B}} \]

      3. *-lft-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F}}{B}} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      6. *-commutative N/A

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

      7. *-lowering-*.f64 8.8

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   10. Simplified 8.8%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.25 \cdot 10^{+193}:\\ \;\;\;\;-\sqrt{F} \cdot \sqrt{\frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 5e+109)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* -2.0 (/ (sqrt (* C F)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5e+109) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 5d+109) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5e+109) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 5e+109:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 5e+109)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 5e+109)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 5e+109], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.0000000000000001e109

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   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 15.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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 16.0

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   7. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]

   if 5.0000000000000001e109 < C

   1. Initial program 6.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 32.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 32.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{{\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}}}{-1} \cdot \frac{{C}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   7. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot 4}}{-1} \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{1 \cdot \sqrt{C \cdot F}}{B}} \]

      3. *-lft-identity N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot F}}{B}} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot F}}}{B} \]

      6. *-commutative N/A

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

      7. *-lowering-*.f64 8.5

         \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{F \cdot C}}}{B} \]

   10. Simplified 8.5%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5 \cdot 10^{+109}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 27.0% 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 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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 13.8

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 13.8%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. *-lowering-*.f64 13.9

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Add Preprocessing

Alternative 21: 27.0% 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 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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 13.8

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 13.8%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   6. *-lowering-*.f64 13.9

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
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 13.9

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 13.9%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))