ABCF->ab-angle a

Percentage Accurate: 18.7% → 57.8%

Time: 20.2s

Alternatives: 16

Speedup: 16.9×

• could not determine a ground truth

  1. A = -7.408433039597928e+186
  2. B = 3.1843883181865133e-15
  3. C = -1.242824044365173e-216
  4. F = 1.2949620460408216e+298

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \frac{\sqrt{4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\right)} \cdot \sqrt{C}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4
         (/
          (* (sqrt (* 4.0 (* F (fma A (* C -4.0) (* B_m B_m))))) (sqrt C))
          t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-228)
       (*
        (/ (sqrt (* 2.0 (* F t_0))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_3 0.0)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= t_3 INFINITY)
           t_4
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = (sqrt((4.0 * (F * fma(A, (C * -4.0), (B_m * B_m))))) * sqrt(C)) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-228) {
		tmp = (sqrt((2.0 * (F * t_0))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_3 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(Float64(sqrt(Float64(4.0 * Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m))))) * sqrt(C)) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-228)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) / -1.0) * Float64(sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) / t_0));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(4.0 * N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-228], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -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 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.7

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

   5. Simplified 27.7%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

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

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

   4. Applied egg-rr 98.3%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1
         (/
          (* (sqrt C) (* (sqrt (* F (fma A (* C -4.0) (* B_m B_m)))) -2.0))
          (fma (* C -4.0) A (* B_m B_m))))
        (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))
     t_1
     (if (<= t_3 -1e-228)
       (*
        (/ (sqrt (* 2.0 (* F t_0))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_3 0.0)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= t_3 INFINITY)
           t_1
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (sqrt(C) * (sqrt((F * fma(A, (C * -4.0), (B_m * B_m)))) * -2.0)) / fma((C * -4.0), A, (B_m * B_m));
	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 = t_1;
	} else if (t_3 <= -1e-228) {
		tmp = (sqrt((2.0 * (F * t_0))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_3 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(sqrt(C) * Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))) * -2.0)) / fma(Float64(C * -4.0), A, Float64(B_m * B_m)))
	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 = t_1;
	elseif (t_3 <= -1e-228)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) / -1.0) * Float64(sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) / t_0));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[C], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $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)], t$95$1, If[LessEqual[t$95$3, -1e-228], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -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 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 27.7

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

   5. Simplified 27.7%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.8%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 38.8

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

   9. Applied egg-rr 38.8%

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

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

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

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

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-neg-in N/A

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

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

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

       8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       14. metadata-eval N/A

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

       15. sqrt-lowering-sqrt.f64 38.8

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

   11. Applied egg-rr 38.8%

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

   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))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. unpow-prod-down N/A

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

      5. neg-mul-1 N/A

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

      6. times-frac N/A

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

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

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

   4. Applied egg-rr 98.3%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 38.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1
         (/
          (* (sqrt C) (* (sqrt (* F (fma A (* C -4.0) (* B_m B_m)))) -2.0))
          (fma (* C -4.0) A (* B_m B_m))))
        (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 -5e+222)
     t_1
     (if (<= t_3 -1e-228)
       (/
        (sqrt
         (*
          (* 2.0 (* F t_0))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_0))
       (if (<= t_3 0.0)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= t_3 INFINITY)
           t_1
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (sqrt(C) * (sqrt((F * fma(A, (C * -4.0), (B_m * B_m)))) * -2.0)) / fma((C * -4.0), A, (B_m * B_m));
	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 <= -5e+222) {
		tmp = t_1;
	} else if (t_3 <= -1e-228) {
		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 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(sqrt(C) * Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))) * -2.0)) / fma(Float64(C * -4.0), A, Float64(B_m * B_m)))
	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 <= -5e+222)
		tmp = t_1;
	elseif (t_3 <= -1e-228)
		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 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[C], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $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, -5e+222], t$95$1, If[LessEqual[t$95$3, -1e-228], 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, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000023e222 or -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 12.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 27.4

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

   5. Simplified 27.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.3%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 38.3

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

   9. Applied egg-rr 38.3%

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

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

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

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

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-neg-in N/A

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

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

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

       8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       14. metadata-eval N/A

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

       15. sqrt-lowering-sqrt.f64 38.3

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

   11. Applied egg-rr 38.3%

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

   if -5.00000000000000023e222 < (/.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))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

      1. 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 98.3%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := \frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)}\\ t_3 := -\sqrt{2}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;t\_3 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0))))
        (t_2
         (/
          (* (sqrt C) (* (sqrt (* F (fma A (* C -4.0) (* B_m B_m)))) -2.0))
          (fma (* C -4.0) A (* B_m B_m))))
        (t_3 (- (sqrt 2.0))))
   (if (<= t_1 -5e+222)
     t_2
     (if (<= t_1 -1e-228)
       (*
        t_3
        (sqrt
         (/
          (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
          (fma B_m B_m (* -4.0 (* A C))))))
       (if (<= t_1 0.0)
         (* (sqrt (/ (* F -0.5) A)) t_3)
         (if (<= t_1 INFINITY)
           t_2
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double t_2 = (sqrt(C) * (sqrt((F * fma(A, (C * -4.0), (B_m * B_m)))) * -2.0)) / fma((C * -4.0), A, (B_m * B_m));
	double t_3 = -sqrt(2.0);
	double tmp;
	if (t_1 <= -5e+222) {
		tmp = t_2;
	} else if (t_1 <= -1e-228) {
		tmp = t_3 * sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else if (t_1 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * t_3;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = Float64(Float64(sqrt(C) * Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))) * -2.0)) / fma(Float64(C * -4.0), A, Float64(B_m * B_m)))
	t_3 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (t_1 <= -5e+222)
		tmp = t_2;
	elseif (t_1 <= -1e-228)
		tmp = Float64(t_3 * 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))))));
	elseif (t_1 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * t_3);
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[C], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[t$95$1, -5e+222], t$95$2, If[LessEqual[t$95$1, -1e-228], N[(t$95$3 * 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]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000023e222 or -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 12.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 27.4

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

   5. Simplified 27.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.3%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 38.3

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

   9. Applied egg-rr 38.3%

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

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

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

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

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-neg-in N/A

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

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

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

       8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       14. metadata-eval N/A

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

       15. sqrt-lowering-sqrt.f64 38.3

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

   11. Applied egg-rr 38.3%

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

   if -5.00000000000000023e222 < (/.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))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

   3. Taylor expanded in 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 96.6%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B\_m \cdot B\_m\right)}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (/
          (* (sqrt C) (* (sqrt (* F (fma A (* C -4.0) (* B_m B_m)))) -2.0))
          (fma (* C -4.0) A (* B_m B_m))))
        (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 -5e+222)
     t_0
     (if (<= t_2 -1e-228)
       (*
        (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))
        (/ (sqrt 2.0) (- B_m)))
       (if (<= t_2 0.0)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= t_2 INFINITY)
           t_0
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (sqrt(C) * (sqrt((F * fma(A, (C * -4.0), (B_m * B_m)))) * -2.0)) / fma((C * -4.0), A, (B_m * B_m));
	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 <= -5e+222) {
		tmp = t_0;
	} else if (t_2 <= -1e-228) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else if (t_2 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(sqrt(C) * Float64(sqrt(Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))) * -2.0)) / fma(Float64(C * -4.0), A, Float64(B_m * B_m)))
	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 <= -5e+222)
		tmp = t_0;
	elseif (t_2 <= -1e-228)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (t_2 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[C], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(C * -4.0), $MachinePrecision] * A + N[(B$95$m * B$95$m), $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, -5e+222], t$95$0, If[LessEqual[t$95$2, -1e-228], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$0, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000023e222 or -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 12.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 27.4

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

   5. Simplified 27.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.3%

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

      1. pow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f64 38.3

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

   9. Applied egg-rr 38.3%

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

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

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

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

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. metadata-eval N/A

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

       6. distribute-rgt-neg-in N/A

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

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

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

       8. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       14. metadata-eval N/A

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

       15. sqrt-lowering-sqrt.f64 38.3

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

   11. Applied egg-rr 38.3%

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

   if -5.00000000000000023e222 < (/.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))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. 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. neg-lowering-neg.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 30.5

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

   7. Simplified 30.5%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{C} \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(\sqrt{F \cdot t\_0} \cdot -2\right) \cdot \frac{\sqrt{C}}{t\_0}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (* (* (sqrt (* F t_0)) -2.0) (/ (sqrt C) 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 -5e+222)
     t_1
     (if (<= t_3 -1e-228)
       (*
        (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))
        (/ (sqrt 2.0) (- B_m)))
       (if (<= t_3 0.0)
         (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
         (if (<= t_3 INFINITY)
           t_1
           (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (sqrt((F * t_0)) * -2.0) * (sqrt(C) / 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 <= -5e+222) {
		tmp = t_1;
	} else if (t_3 <= -1e-228) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else if (t_3 <= 0.0) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(sqrt(Float64(F * t_0)) * -2.0) * Float64(sqrt(C) / 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 <= -5e+222)
		tmp = t_1;
	elseif (t_3 <= -1e-228)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[Sqrt[C], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+222], t$95$1, If[LessEqual[t$95$3, -1e-228], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$1, N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000023e222 or -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 12.9%

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

   3. Taylor expanded in A around -inf

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

      1. *-lowering-*.f64 27.4

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

   5. Simplified 27.4%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 38.3%

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

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

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

      2. pow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

   9. Applied egg-rr 38.2%

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

   if -5.00000000000000023e222 < (/.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))) < -1.00000000000000003e-228

   1. Initial program 98.3%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 93.6%

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

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. 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. neg-lowering-neg.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 30.5

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

   7. Simplified 30.5%

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

   if -1.00000000000000003e-228 < (/.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 3.5%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 18.8%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.4

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

   8. Simplified 31.4%

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

   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 14.5

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

   5. Simplified 14.5%

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

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

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

      2. sqrt-unprod N/A

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

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

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

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

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

      5. /-lowering-/.f64 14.5

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

   7. Applied egg-rr 14.5%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 14.5

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

   9. Applied egg-rr 14.5%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 19.9

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

   11. Applied egg-rr 19.9%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+222}:\\ \;\;\;\;\left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\left(\sqrt{F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot -2\right) \cdot \frac{\sqrt{C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if ((Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_0 - Math.pow(B_m, 2.0))) <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 * ((1.0 / B_m) * Math.sqrt((C * F)));
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if (math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_0 - math.pow(B_m, 2.0))) <= -math.inf:
		tmp = -2.0 * ((1.0 / B_m) * math.sqrt((C * F)))
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0))) <= Float64(-Inf))
		tmp = Float64(-2.0 * Float64(Float64(1.0 / B_m) * sqrt(Float64(C * F))));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_0 - (B_m ^ 2.0))) <= -Inf)
		tmp = -2.0 * ((1.0 / B_m) * sqrt((C * F)));
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.0%

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

   3. Taylor expanded in 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 20.7

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

   5. Simplified 20.7%

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

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. 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. *-lowering-*.f64 N/A

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

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

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

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

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

      5. *-lowering-*.f64 7.6

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

   8. Simplified 7.6%

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

   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 21.8%

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

   3. Taylor expanded in 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 13.3

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

   5. Simplified 13.3%

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

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

      5. /-lowering-/.f64 13.3

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

   7. Applied egg-rr 13.3%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 13.3

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

   9. Applied egg-rr 13.3%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. div-inv N/A

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

       4. associate-/r/ N/A

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

       5. sqrt-prod N/A

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

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

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

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

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

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

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

       9. sqrt-lowering-sqrt.f64 16.3

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

   11. Applied egg-rr 16.3%

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

4. Final simplification 14.5%

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

Alternative 8: 49.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B\_m, B\_m, t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C))))
   (if (<= (pow B_m 2.0) 5e-242)
     (/ (* -2.0 (sqrt (* (* C F) (fma B_m B_m t_0)))) t_0)
     (if (<= (pow B_m 2.0) 1.8e-110)
       (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 1e+184)
         (*
          (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))
          (/ (sqrt 2.0) (- B_m)))
         (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-242) {
		tmp = (-2.0 * sqrt(((C * F) * fma(B_m, B_m, t_0)))) / t_0;
	} else if (pow(B_m, 2.0) <= 1.8e-110) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 1e+184) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-242)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(C * F) * fma(B_m, B_m, t_0)))) / t_0);
	elseif ((B_m ^ 2.0) <= 1.8e-110)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 1e+184)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-242], N[(N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.8e-110], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+184], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e-242

   1. Initial program 18.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.2

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

   5. Simplified 31.2%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 31.3%

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

   8. Taylor expanded in B around 0

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

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

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

      2. *-lowering-*.f64 31.2

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

   10. Simplified 31.2%

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

   11. Taylor expanded in F around 0

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

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

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

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

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

       3. associate-*r* N/A

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

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

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

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

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

       6. +-commutative N/A

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

       7. unpow2 N/A

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

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

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

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

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

       10. *-lowering-*.f64 31.1

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

   13. Simplified 31.1%

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

   if 4.9999999999999998e-242 < (pow.f64 B #s(literal 2 binary64)) < 1.79999999999999997e-110

   1. Initial program 16.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 28.9%

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

   6. Taylor expanded in C around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 26.0

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

   8. Simplified 26.0%

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

   if 1.79999999999999997e-110 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e184

   1. Initial program 33.4%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 34.7%

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

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. 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. neg-lowering-neg.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 13.9

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

   7. Simplified 13.9%

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

   if 1.00000000000000002e184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.9

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

   5. Simplified 20.9%

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

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

      5. /-lowering-/.f64 20.9

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

   7. Applied egg-rr 20.9%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 20.9

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

   9. Applied egg-rr 20.9%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 29.2

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

   11. Applied egg-rr 29.2%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(t\_0 \cdot \left(4 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-64)
     (/ (sqrt (* C (* t_0 (* 4.0 F)))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+184)
       (*
        (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))
        (/ (sqrt 2.0) (- B_m)))
       (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-64) {
		tmp = sqrt((C * (t_0 * (4.0 * F)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+184) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-64)
		tmp = Float64(sqrt(Float64(C * Float64(t_0 * Float64(4.0 * F)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+184)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-64], N[(N[Sqrt[N[(C * N[(t$95$0 * N[(4.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+184], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000033e-64

   1. Initial program 18.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.7

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

   5. Simplified 25.7%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. unpow-prod-down N/A

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

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

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

   7. Applied egg-rr 24.1%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   9. Applied egg-rr 25.7%

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

   if 5.00000000000000033e-64 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e184

   1. Initial program 36.0%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

      3. pow1/2 N/A

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

      4. associate-*l* N/A

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

      5. unpow-prod-down N/A

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

      6. neg-mul-1 N/A

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

      7. times-frac N/A

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

   4. Applied egg-rr 35.9%

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

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. 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. neg-lowering-neg.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 15.3

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

   7. Simplified 15.3%

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

   if 1.00000000000000002e184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.9

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

   5. Simplified 20.9%

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

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

      5. /-lowering-/.f64 20.9

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

   7. Applied egg-rr 20.9%

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-lowering-*.f64 20.9

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

   9. Applied egg-rr 20.9%

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

       1. sqrt-div N/A

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

       2. distribute-neg-frac N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

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

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}}} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot F}}\right)}} \]

       9. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}}\right)}} \]

       10. *-lowering-*.f64 29.2

           \[\leadsto \frac{1}{\frac{\sqrt{B}}{-\sqrt{\color{blue}{F \cdot 2}}}} \]

   11. Applied egg-rr 29.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{B}}{-\sqrt{F \cdot 2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(4 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{2 \cdot F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-64)
     (/ (sqrt (* C (* 4.0 (* F t_0)))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+184)
       (*
        (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))
        (/ (sqrt 2.0) (- B_m)))
       (/ -1.0 (/ (sqrt B_m) (sqrt (* 2.0 F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-64) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+184) {
		tmp = sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -1.0 / (sqrt(B_m) / sqrt((2.0 * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-64)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+184)
		tmp = Float64(sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) / sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-64], 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+184], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000033e-64

   1. Initial program 18.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 25.7

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 25.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. pow2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left(B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}\right)\right)} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\color{blue}{\left(B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right)}\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{neg}\left(\left(-4 \cdot \left(A \cdot C\right) + B \cdot B\right)\right)}} \]

   7. Applied egg-rr 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 5.00000000000000033e-64 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e184

   1. Initial program 36.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{{\color{blue}{\left(2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}^{\frac{1}{2}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{\color{blue}{{2}^{\frac{1}{2}} \cdot {\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      6. neg-mul-1 N/A

         \[\leadsto \frac{{2}^{\frac{1}{2}} \cdot {\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}{\color{blue}{-1 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

      7. times-frac N/A

         \[\leadsto \color{blue}{\frac{{2}^{\frac{1}{2}}}{-1} \cdot \frac{{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   4. Applied egg-rr 35.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{-1} \cdot \frac{\sqrt{F \cdot \left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]

   5. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   6. 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. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \color{blue}{\sqrt{{B}^{2} + {C}^{2}}}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {C}^{2}\right)}}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)}\right) \]

      13. *-lowering-*.f64 15.3

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)} \]

   7. Simplified 15.3%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}} \]

   if 1.00000000000000002e184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 3.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 20.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 20.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      5. /-lowering-/.f64 20.9

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

   7. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
   8. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. *-lowering-*.f64 20.9

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   9. Applied egg-rr 20.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
   10. Step-by-step derivation

       1. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2 \cdot F}}{\sqrt{B}}}\right) \]

       2. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}{\sqrt{B}}} \]

       3. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{B}}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}}} \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{B}}}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}} \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot F}\right)}}} \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot F}}\right)}} \]

       9. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\sqrt{B}}{\mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}}\right)}} \]

       10. *-lowering-*.f64 29.2

           \[\leadsto \frac{1}{\frac{\sqrt{B}}{-\sqrt{\color{blue}{F \cdot 2}}}} \]

   11. Applied egg-rr 29.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{B}}{-\sqrt{F \cdot 2}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+184}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{B}}{\sqrt{2 \cdot F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B\_m, B\_m, t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C))))
   (if (<= (pow B_m 2.0) 5e-242)
     (/ (* -2.0 (sqrt (* (* C F) (fma B_m B_m t_0)))) t_0)
     (if (<= (pow B_m 2.0) 1e+38)
       (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
       (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double tmp;
	if (pow(B_m, 2.0) <= 5e-242) {
		tmp = (-2.0 * sqrt(((C * F) * fma(B_m, B_m, t_0)))) / t_0;
	} else if (pow(B_m, 2.0) <= 1e+38) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-242)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(C * F) * fma(B_m, B_m, t_0)))) / t_0);
	elseif ((B_m ^ 2.0) <= 1e+38)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-242], N[(N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+38], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e-242

   1. Initial program 18.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 31.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied egg-rr 31.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in B around 0

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

      2. *-lowering-*.f64 31.2

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   10. Simplified 31.2%

       \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   11. Taylor expanded in F around 0

       \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       3. associate-*r* N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right)} \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)}}{-4 \cdot \left(A \cdot C\right)} \]

       6. +-commutative N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       7. unpow2 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \left(A \cdot C\right)} \]

       8. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{-4 \cdot \left(A \cdot C\right)} \]

       10. *-lowering-*.f64 31.1

           \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{-4 \cdot \left(A \cdot C\right)} \]

   13. Simplified 31.1%

       \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{-4 \cdot \left(A \cdot C\right)} \]

   if 4.9999999999999998e-242 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e37

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 32.9%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   6. Taylor expanded in C around inf

      \[\leadsto \sqrt{\color{blue}{\frac{-1}{2} \cdot \frac{F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 12.3

         \[\leadsto \sqrt{\frac{\color{blue}{-0.5 \cdot F}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 12.3%

      \[\leadsto \sqrt{\color{blue}{\frac{-0.5 \cdot F}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 9.99999999999999977e37 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 18.6

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 18.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      5. /-lowering-/.f64 18.7

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

   7. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
   8. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. *-lowering-*.f64 18.7

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   9. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}}\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}}\right) \]

       4. associate-/r/ N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B} \cdot F}}\right) \]

       5. sqrt-prod N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       9. sqrt-lowering-sqrt.f64 24.4

          \[\leadsto -\sqrt{\frac{2}{B}} \cdot \color{blue}{\sqrt{F}} \]

   11. Applied egg-rr 24.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-242)
   (/
    (sqrt (* C (* 4.0 (* F (fma A (* C -4.0) (* B_m B_m))))))
    (* 4.0 (* A C)))
   (if (<= (pow B_m 2.0) 1e+38)
     (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
     (* (sqrt (/ 2.0 B_m)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-242) {
		tmp = sqrt((C * (4.0 * (F * fma(A, (C * -4.0), (B_m * B_m)))))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 1e+38) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-242)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * fma(A, Float64(C * -4.0), Float64(B_m * B_m)))))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 1e+38)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-242], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+38], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e-242

   1. Initial program 18.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 31.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 31.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)\right)}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left({\color{blue}{\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right) \cdot C\right)}}^{\frac{1}{2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. unpow-prod-down N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot 2\right)}^{\frac{1}{2}} \cdot {C}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied egg-rr 31.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in B around 0

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

      2. *-lowering-*.f64 31.2

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   10. Simplified 31.2%

       \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(\left(A \cdot \left(C \cdot -4\right) + B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}\right)\right)\right)}{\mathsf{neg}\left(-4 \cdot \left(A \cdot C\right)\right)}} \]

       2. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\left(\left(A \cdot \left(C \cdot -4\right) + B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}}{\mathsf{neg}\left(-4 \cdot \left(A \cdot C\right)\right)} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(A \cdot \left(C \cdot -4\right) + B \cdot B\right) \cdot F\right) \cdot 4} \cdot \sqrt{C}}{\mathsf{neg}\left(-4 \cdot \left(A \cdot C\right)\right)}} \]

   12. Applied egg-rr 31.1%

       \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4\right)}}{4 \cdot \left(A \cdot C\right)}} \]

   if 4.9999999999999998e-242 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e37

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   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 32.9%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   6. Taylor expanded in C around inf

      \[\leadsto \sqrt{\color{blue}{\frac{-1}{2} \cdot \frac{F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 12.3

         \[\leadsto \sqrt{\frac{\color{blue}{-0.5 \cdot F}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 12.3%

      \[\leadsto \sqrt{\color{blue}{\frac{-0.5 \cdot F}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 9.99999999999999977e37 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 18.6

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 18.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      5. /-lowering-/.f64 18.7

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

   7. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
   8. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. *-lowering-*.f64 18.7

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   9. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}}\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}}\right) \]

       4. associate-/r/ N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B} \cdot F}}\right) \]

       5. sqrt-prod N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       9. sqrt-lowering-sqrt.f64 24.4

          \[\leadsto -\sqrt{\frac{2}{B}} \cdot \color{blue}{\sqrt{F}} \]

   11. Applied egg-rr 24.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+38)
   (* (sqrt (/ (* F -0.5) A)) (- (sqrt 2.0)))
   (* (sqrt (/ 2.0 B_m)) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+38) {
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if ((b_m ** 2.0d0) <= 1d+38) then
        tmp = sqrt(((f * (-0.5d0)) / a)) * -sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 / b_m)) * -sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+38) {
		tmp = Math.sqrt(((F * -0.5) / A)) * -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 / B_m)) * -Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+38:
		tmp = math.sqrt(((F * -0.5) / A)) * -math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 / B_m)) * -math.sqrt(F)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+38)
		tmp = Float64(sqrt(Float64(Float64(F * -0.5) / A)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(-sqrt(F)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+38)
		tmp = sqrt(((F * -0.5) / A)) * -sqrt(2.0);
	else
		tmp = sqrt((2.0 / B_m)) * -sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+38], N[(N[Sqrt[N[(N[(F * -0.5), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e37

   1. Initial program 21.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right)} \]

   5. Simplified 21.5%

      \[\leadsto \color{blue}{\sqrt{\frac{F \cdot \left(\left(C + A\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)}} \cdot \left(-\sqrt{2}\right)} \]

   6. Taylor expanded in C around inf

      \[\leadsto \sqrt{\color{blue}{\frac{-1}{2} \cdot \frac{F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{2} \cdot F}{A}}} \cdot \left(\mathsf{neg}\left(\sqrt{2}\right)\right) \]

      3. *-lowering-*.f64 18.0

         \[\leadsto \sqrt{\frac{\color{blue}{-0.5 \cdot F}}{A}} \cdot \left(-\sqrt{2}\right) \]

   8. Simplified 18.0%

      \[\leadsto \sqrt{\color{blue}{\frac{-0.5 \cdot F}{A}}} \cdot \left(-\sqrt{2}\right) \]

   if 9.99999999999999977e37 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. /-lowering-/.f64 18.6

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 18.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      5. /-lowering-/.f64 18.7

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

   7. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
   8. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. *-lowering-*.f64 18.7

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   9. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}}\right) \]

       3. div-inv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}}\right) \]

       4. associate-/r/ N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B} \cdot F}}\right) \]

       5. sqrt-prod N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2}{B}}} \cdot \sqrt{F}\right) \]

       9. sqrt-lowering-sqrt.f64 24.4

          \[\leadsto -\sqrt{\frac{2}{B}} \cdot \color{blue}{\sqrt{F}} \]

   11. Applied egg-rr 24.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\frac{F \cdot -0.5}{A}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 6.5 \cdot 10^{+55}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\frac{1}{B\_m} \cdot \sqrt{C \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 6.5e+55)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* -2.0 (* (/ 1.0 B_m) (sqrt (* C F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 6.5e+55) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * ((1.0 / B_m) * sqrt((C * F)));
	}
	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 <= 6.5d+55) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = (-2.0d0) * ((1.0d0 / b_m) * sqrt((c * f)))
    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 <= 6.5e+55) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = -2.0 * ((1.0 / B_m) * Math.sqrt((C * F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 6.5e+55:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = -2.0 * ((1.0 / B_m) * math.sqrt((C * F)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 6.5e+55)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(-2.0 * Float64(Float64(1.0 / B_m) * sqrt(Float64(C * F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 6.5e+55)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = -2.0 * ((1.0 / B_m) * sqrt((C * F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 6.5e+55], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[(1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 6.50000000000000027e55

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 13.3

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 13.3%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      5. /-lowering-/.f64 13.3

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

   7. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
   8. Step-by-step derivation

      1. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      5. *-lowering-*.f64 13.3

         \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]

   if 6.50000000000000027e55 < C

   1. Initial program 17.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 31.7

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 31.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. 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. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \left(\color{blue}{\frac{1}{B}} \cdot \sqrt{C \cdot F}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\frac{1}{B} \cdot \color{blue}{\sqrt{C \cdot F}}\right) \]

      5. *-lowering-*.f64 8.3

         \[\leadsto -2 \cdot \left(\frac{1}{B} \cdot \sqrt{\color{blue}{C \cdot F}}\right) \]

   8. Simplified 8.3%

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B} \cdot \sqrt{C \cdot F}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 26.4% 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 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 11.4

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.4%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   5. /-lowering-/.f64 11.4

      \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

7. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
8. Step-by-step derivation

   1. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B}}\right)} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   5. *-lowering-*.f64 11.4

      \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

9. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
10. Add Preprocessing

Alternative 16: 26.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ F B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((2.0 * (F / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((2.0d0 * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(2.0 * Float64(F / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 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 11.4

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 11.4%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   5. /-lowering-/.f64 11.4

      \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{F}{B}}} \]

7. Applied egg-rr 11.4%

   \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))