ABCF->ab-angle a

Percentage Accurate: 18.7% → 62.1%

Time: 19.9s

Alternatives: 22

Speedup: 16.9×

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: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \sqrt{F}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{t\_1 \cdot \mathsf{fma}\left(2, B\_m \cdot B\_m, \left(A \cdot C\right) \cdot -8\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{t\_1}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 -5e+235)
     (/
      (*
       (sqrt (* 2.0 C))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (* B_m B_m)))) (sqrt F)))
      t_3)
     (if (<= t_4 -1e-220)
       (*
        (/ (sqrt (* t_0 (* 2.0 F))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_4 0.0)
         (/
          (* (sqrt F) (sqrt (* t_1 (fma 2.0 (* B_m B_m) (* (* A C) -8.0)))))
          t_3)
         (if (<= t_4 INFINITY)
           (/
            (* (sqrt (* (fma B_m B_m (* A (* C -4.0))) (* 2.0 F))) (sqrt t_1))
            t_3)
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -5e+235) {
		tmp = (sqrt((2.0 * C)) * (sqrt((2.0 * fma(C, (A * -4.0), (B_m * B_m)))) * sqrt(F))) / t_3;
	} else if (t_4 <= -1e-220) {
		tmp = (sqrt((t_0 * (2.0 * F))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt((t_1 * fma(2.0, (B_m * B_m), ((A * C) * -8.0))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((fma(B_m, B_m, (A * (C * -4.0))) * (2.0 * F))) * sqrt(t_1)) / t_3;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -5e+235)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))) * sqrt(F))) / t_3);
	elseif (t_4 <= -1e-220)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / -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_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(t_1 * fma(2.0, Float64(B_m * B_m), Float64(Float64(A * C) * -8.0))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) * Float64(2.0 * F))) * sqrt(t_1)) / t_3);
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+235], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-220], N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $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$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000027e235

   1. Initial program 6.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   9. Applied rewrites 16.4%

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

   if -5.00000000000000027e235 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 27.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

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

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

   7. Applied rewrites 37.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\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:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(2, B \cdot B, \left(A \cdot C\right) \cdot -8\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \sqrt{F}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{t\_1 \cdot \mathsf{fma}\left(2, B\_m \cdot B\_m, \left(A \cdot C\right) \cdot -8\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 -5e+235)
     (/
      (*
       (sqrt (* 2.0 C))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (* B_m B_m)))) (sqrt F)))
      t_3)
     (if (<= t_4 -1e-220)
       (*
        (/ (sqrt (* t_0 (* 2.0 F))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_4 0.0)
         (/
          (* (sqrt F) (sqrt (* t_1 (fma 2.0 (* B_m B_m) (* (* A C) -8.0)))))
          t_3)
         (if (<= t_4 INFINITY)
           (/
            (* (sqrt (* 2.0 t_1)) (sqrt (* F (fma B_m B_m (* A (* C -4.0))))))
            t_3)
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -5e+235) {
		tmp = (sqrt((2.0 * C)) * (sqrt((2.0 * fma(C, (A * -4.0), (B_m * B_m)))) * sqrt(F))) / t_3;
	} else if (t_4 <= -1e-220) {
		tmp = (sqrt((t_0 * (2.0 * F))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) * sqrt((t_1 * fma(2.0, (B_m * B_m), ((A * C) * -8.0))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_1)) * sqrt((F * fma(B_m, B_m, (A * (C * -4.0)))))) / t_3;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -5e+235)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))) * sqrt(F))) / t_3);
	elseif (t_4 <= -1e-220)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / -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_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(t_1 * fma(2.0, Float64(B_m * B_m), Float64(Float64(A * C) * -8.0))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))))) / t_3);
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+235], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-220], N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $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$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000027e235

   1. Initial program 6.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   9. Applied rewrites 16.4%

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

   if -5.00000000000000027e235 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 27.2%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 37.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\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:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(2, B \cdot B, \left(A \cdot C\right) \cdot -8\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := 2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \sqrt{F}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{t\_5}{\sqrt{F} \cdot \sqrt{t\_1 \cdot \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \sqrt{F \cdot t\_5}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (* 2.0 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3))
        (t_5 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= t_4 -5e+235)
     (/
      (*
       (sqrt (* 2.0 C))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (* B_m B_m)))) (sqrt F)))
      t_3)
     (if (<= t_4 -1e-220)
       (*
        (/ (sqrt (* t_0 (* 2.0 F))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_4 0.0)
         (/
          -1.0
          (/ t_5 (* (sqrt F) (sqrt (* t_1 (fma -4.0 (* A C) (* B_m B_m)))))))
         (if (<= t_4 INFINITY)
           (/ (* (sqrt t_1) (sqrt (* F t_5))) t_3)
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = 2.0 * fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (t_4 <= -5e+235) {
		tmp = (sqrt((2.0 * C)) * (sqrt((2.0 * fma(C, (A * -4.0), (B_m * B_m)))) * sqrt(F))) / t_3;
	} else if (t_4 <= -1e-220) {
		tmp = (sqrt((t_0 * (2.0 * F))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_4 <= 0.0) {
		tmp = -1.0 / (t_5 / (sqrt(F) * sqrt((t_1 * fma(-4.0, (A * C), (B_m * B_m))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(t_1) * sqrt((F * t_5))) / t_3;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(2.0 * fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	t_5 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_4 <= -5e+235)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))) * sqrt(F))) / t_3);
	elseif (t_4 <= -1e-220)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / -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_4 <= 0.0)
		tmp = Float64(-1.0 / Float64(t_5 / Float64(sqrt(F) * sqrt(Float64(t_1 * fma(-4.0, Float64(A * C), Float64(B_m * B_m)))))));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(t_1) * sqrt(Float64(F * t_5))) / t_3);
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+235], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-220], N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $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$4, 0.0], N[(-1.0 / N[(t$95$5 / N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(F * t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000027e235

   1. Initial program 6.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   9. Applied rewrites 16.4%

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

   if -5.00000000000000027e235 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 22.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   9. Applied rewrites 27.3%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 37.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\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:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ t_5 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)} \cdot \sqrt{F}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F} \cdot \sqrt{\left(2 \cdot t\_5\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{t\_5}}{\mathsf{fma}\left(B\_m, -B\_m, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (fma -4.0 (* A C) (* B_m B_m)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3))
        (t_5 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A))))
   (if (<= t_4 -5e+235)
     (/
      (*
       (sqrt (* 2.0 C))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (* B_m B_m)))) (sqrt F)))
      t_3)
     (if (<= t_4 -1e-220)
       (*
        (/ (sqrt (* t_0 (* 2.0 F))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_0))
       (if (<= t_4 0.0)
         (/
          -1.0
          (/
           (fma B_m B_m (* A (* C -4.0)))
           (* (sqrt F) (sqrt (* (* 2.0 t_5) t_1)))))
         (if (<= t_4 INFINITY)
           (*
            (sqrt (* t_1 (* 2.0 F)))
            (/ (sqrt t_5) (fma B_m (- B_m) (* 4.0 (* A C)))))
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = fma(-4.0, (A * C), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double tmp;
	if (t_4 <= -5e+235) {
		tmp = (sqrt((2.0 * C)) * (sqrt((2.0 * fma(C, (A * -4.0), (B_m * B_m)))) * sqrt(F))) / t_3;
	} else if (t_4 <= -1e-220) {
		tmp = (sqrt((t_0 * (2.0 * F))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_0);
	} else if (t_4 <= 0.0) {
		tmp = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / (sqrt(F) * sqrt(((2.0 * t_5) * t_1))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (2.0 * F))) * (sqrt(t_5) / fma(B_m, -B_m, (4.0 * (A * C))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	t_5 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	tmp = 0.0
	if (t_4 <= -5e+235)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))) * sqrt(F))) / t_3);
	elseif (t_4 <= -1e-220)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / -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_4 <= 0.0)
		tmp = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * t_5) * t_1)))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * F))) * Float64(sqrt(t_5) / fma(B_m, Float64(-B_m), Float64(4.0 * Float64(A * C)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+235], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-220], N[(N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $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$4, 0.0], N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$5], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000027e235

   1. Initial program 6.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 22.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.2

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

   7. Applied rewrites 14.2%

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

   9. Applied rewrites 16.4%

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

   if -5.00000000000000027e235 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 22.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   9. Applied rewrites 27.3%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 37.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\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:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_4 \cdot t\_1} \cdot \sqrt{2 \cdot F}}}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_5}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{F} \cdot \sqrt{\left(2 \cdot t\_4\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{t\_4}}{\mathsf{fma}\left(B\_m, -B\_m, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (fma -4.0 (* A C) (* 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))))
        (t_4 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
        (t_5 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_3 -5e+235)
     (/ -1.0 (/ t_0 (* (sqrt (* t_4 t_1)) (sqrt (* 2.0 F)))))
     (if (<= t_3 -1e-220)
       (*
        (/ (sqrt (* t_5 (* 2.0 F))) -1.0)
        (/ (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))) t_5))
       (if (<= t_3 0.0)
         (/ -1.0 (/ t_0 (* (sqrt F) (sqrt (* (* 2.0 t_4) t_1)))))
         (if (<= t_3 INFINITY)
           (*
            (sqrt (* t_1 (* 2.0 F)))
            (/ (sqrt t_4) (fma B_m (- B_m) (* 4.0 (* A C)))))
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(-4.0, (A * C), (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 t_4 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_5 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_3 <= -5e+235) {
		tmp = -1.0 / (t_0 / (sqrt((t_4 * t_1)) * sqrt((2.0 * F))));
	} else if (t_3 <= -1e-220) {
		tmp = (sqrt((t_5 * (2.0 * F))) / -1.0) * (sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m))))) / t_5);
	} else if (t_3 <= 0.0) {
		tmp = -1.0 / (t_0 / (sqrt(F) * sqrt(((2.0 * t_4) * t_1))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (2.0 * F))) * (sqrt(t_4) / fma(B_m, -B_m, (4.0 * (A * C))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(-4.0, Float64(A * C), 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)))
	t_4 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	t_5 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_3 <= -5e+235)
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(Float64(t_4 * t_1)) * sqrt(Float64(2.0 * F)))));
	elseif (t_3 <= -1e-220)
		tmp = Float64(Float64(sqrt(Float64(t_5 * Float64(2.0 * F))) / -1.0) * Float64(sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))) / t_5));
	elseif (t_3 <= 0.0)
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * t_4) * t_1)))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * F))) * Float64(sqrt(t_4) / fma(B_m, Float64(-B_m), Float64(4.0 * Float64(A * C)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+235], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[N[(t$95$4 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-220], N[(N[(N[Sqrt[N[(t$95$5 * N[(2.0 * F), $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$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000027e235

   1. Initial program 6.0%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 8.5

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

   5. Applied rewrites 8.5%

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

   7. Applied rewrites 8.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

   9. Applied rewrites 17.7%

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

   if -5.00000000000000027e235 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 22.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   9. Applied rewrites 27.3%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 37.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+235}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{2 \cdot F}}}\\ \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^{-220}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-1} \cdot \frac{\sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\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:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_4 \cdot t\_1} \cdot \sqrt{2 \cdot F}}}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(t\_5 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_5}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{F} \cdot \sqrt{\left(2 \cdot t\_4\right) \cdot t\_1}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{t\_4}}{\mathsf{fma}\left(B\_m, -B\_m, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (fma -4.0 (* A C) (* 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))))
        (t_4 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
        (t_5 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_3 (- INFINITY))
     (/ -1.0 (/ t_0 (* (sqrt (* t_4 t_1)) (sqrt (* 2.0 F)))))
     (if (<= t_3 -1e-220)
       (*
        (sqrt
         (*
          (* t_5 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_5))
       (if (<= t_3 0.0)
         (/ -1.0 (/ t_0 (* (sqrt F) (sqrt (* (* 2.0 t_4) t_1)))))
         (if (<= t_3 INFINITY)
           (*
            (sqrt (* t_1 (* 2.0 F)))
            (/ (sqrt t_4) (fma B_m (- B_m) (* 4.0 (* A C)))))
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = fma(-4.0, (A * C), (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 t_4 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_5 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 / (t_0 / (sqrt((t_4 * t_1)) * sqrt((2.0 * F))));
	} else if (t_3 <= -1e-220) {
		tmp = sqrt(((t_5 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_5);
	} else if (t_3 <= 0.0) {
		tmp = -1.0 / (t_0 / (sqrt(F) * sqrt(((2.0 * t_4) * t_1))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (2.0 * F))) * (sqrt(t_4) / fma(B_m, -B_m, (4.0 * (A * C))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = fma(-4.0, Float64(A * C), 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)))
	t_4 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	t_5 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(Float64(t_4 * t_1)) * sqrt(Float64(2.0 * F)))));
	elseif (t_3 <= -1e-220)
		tmp = Float64(sqrt(Float64(Float64(t_5 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_5));
	elseif (t_3 <= 0.0)
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * t_4) * t_1)))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * F))) * Float64(sqrt(t_4) / fma(B_m, Float64(-B_m), Float64(4.0 * Float64(A * C)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[N[(t$95$4 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-220], N[(N[Sqrt[N[(N[(t$95$5 * N[(2.0 * F), $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] * N[(-1.0 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$4], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.6%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 6.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

   9. Applied rewrites 15.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 3.8%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 22.3

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

   5. Applied rewrites 22.3%

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

   7. Applied rewrites 22.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

   9. Applied rewrites 27.3%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 37.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{2 \cdot F}}}\\ \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^{-220}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\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:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\\ t_4 := \frac{-1}{\frac{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{t\_3 \cdot t\_0} \cdot \sqrt{2 \cdot F}}}\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(t\_5 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_5}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{t\_3}}{\mathsf{fma}\left(B\_m, -B\_m, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* 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))))
        (t_3 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
        (t_4
         (/
          -1.0
          (/
           (fma B_m B_m (* A (* C -4.0)))
           (* (sqrt (* t_3 t_0)) (sqrt (* 2.0 F))))))
        (t_5 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 -1e-220)
       (*
        (sqrt
         (*
          (* t_5 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_5))
       (if (<= t_2 0.0)
         t_4
         (if (<= t_2 INFINITY)
           (*
            (sqrt (* t_0 (* 2.0 F)))
            (/ (sqrt t_3) (fma B_m (- B_m) (* 4.0 (* A C)))))
           (* (- (sqrt F)) (sqrt (/ 2.0 B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (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 t_3 = fma(2.0, C, (((B_m * B_m) * -0.5) / A));
	double t_4 = -1.0 / (fma(B_m, B_m, (A * (C * -4.0))) / (sqrt((t_3 * t_0)) * sqrt((2.0 * F))));
	double t_5 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= -1e-220) {
		tmp = sqrt(((t_5 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_5);
	} else if (t_2 <= 0.0) {
		tmp = t_4;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_0 * (2.0 * F))) * (sqrt(t_3) / fma(B_m, -B_m, (4.0 * (A * C))));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), 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)))
	t_3 = fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))
	t_4 = Float64(-1.0 / Float64(fma(B_m, B_m, Float64(A * Float64(C * -4.0))) / Float64(sqrt(Float64(t_3 * t_0)) * sqrt(Float64(2.0 * F)))))
	t_5 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(Float64(t_5 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_5));
	elseif (t_2 <= 0.0)
		tmp = t_4;
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) * Float64(sqrt(t_3) / fma(B_m, Float64(-B_m), Float64(4.0 * Float64(A * C)))));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-1.0 / N[(N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(t$95$3 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(N[(t$95$5 * N[(2.0 * F), $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] * N[(-1.0 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$4, If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$3], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -9.99999999999999992e-221 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 13.8

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

   5. Applied rewrites 13.8%

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

   7. Applied rewrites 13.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

   9. Applied rewrites 21.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 48.9%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 37.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{2 \cdot F}}}\\ \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^{-220}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\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:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{2 \cdot F}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, 4 \cdot \left(A \cdot C\right)\right)}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(t\_3 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (* (fma -4.0 (* A C) (* B_m B_m)) (* 2.0 F)))
          (/
           (sqrt (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))
           (fma B_m (- B_m) (* 4.0 (* A C))))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_2 (- INFINITY))
     t_0
     (if (<= t_2 -1e-220)
       (*
        (sqrt
         (*
          (* t_3 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_3))
       (if (<= t_2 INFINITY) t_0 (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((fma(-4.0, (A * C), (B_m * B_m)) * (2.0 * F))) * (sqrt(fma(2.0, C, (((B_m * B_m) * -0.5) / A))) / fma(B_m, -B_m, (4.0 * (A * C))));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= -1e-220) {
		tmp = sqrt(((t_3 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) * Float64(2.0 * F))) * Float64(sqrt(fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))) / fma(B_m, Float64(-B_m), Float64(4.0 * Float64(A * C)))))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(Float64(t_3 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_3));
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(N[(t$95$3 * N[(2.0 * F), $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] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$0, N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -9.99999999999999992e-221 < (/.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 12.2%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 17.1

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

   5. Applied rewrites 17.1%

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

   7. Applied rewrites 17.0%

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

   9. Applied rewrites 19.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)}}{\mathsf{fma}\left(B, -B, 4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(t\_3 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_2 (- INFINITY))
     (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
     (if (<= t_2 -1e-220)
       (*
        (sqrt
         (*
          (* t_3 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_3))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)) (* t_0 (* 2.0 F))))
          (- t_0))
         (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else if (t_2 <= -1e-220) {
		tmp = sqrt(((t_3 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(2.0, C, (((B_m * B_m) * -0.5) / A)) * (t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(Float64(t_3 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(N[(t$95$3 * N[(2.0 * F), $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] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $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

   1. Initial program 3.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 8.4%

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

   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))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   4. Applied rewrites 98.8%

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

   if -9.99999999999999992e-221 < (/.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 18.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 25.3

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 25.3%

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

   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. lower-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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\left(t\_3 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_2 (- INFINITY))
     (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
     (if (<= t_2 -1e-220)
       (/
        (sqrt
         (*
          (* t_3 (* 2.0 F))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (- t_3))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)) (* t_0 (* 2.0 F))))
          (- t_0))
         (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else if (t_2 <= -1e-220) {
		tmp = sqrt(((t_3 * (2.0 * F)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / -t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((fma(2.0, C, (((B_m * B_m) * -0.5) / A)) * (t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(Float64(t_3 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / Float64(-t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(N[(t$95$3 * N[(2.0 * F), $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$3)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $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

   1. Initial program 3.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 8.4%

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

   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))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.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 rewrites 98.5%

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

   if -9.99999999999999992e-221 < (/.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 18.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 25.3

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

   5. Applied rewrites 25.3%

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

   7. Applied rewrites 25.3%

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

   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. lower-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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B\_m \cdot B\_m\right)\right)}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) (- 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 (- INFINITY))
     (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) t_0)
     (if (<= t_2 -1e-220)
       (/ -1.0 (/ B_m (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C)))))))))
       (if (<= t_2 INFINITY)
         (* t_0 (sqrt (/ (* -0.5 (* F (* B_m B_m))) A)))
         (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / -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 <= -((double) INFINITY)) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * t_0;
	} else if (t_2 <= -1e-220) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * sqrt(((-0.5 * (F * (B_m * B_m))) / A));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / Float64(-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 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * t_0);
	elseif (t_2 <= -1e-220)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))))));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * sqrt(Float64(Float64(-0.5 * Float64(F * Float64(B_m * B_m))) / A)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $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, (-Infinity)], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e-220], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[Sqrt[N[(N[(-0.5 * N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $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

   1. Initial program 3.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 8.4%

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

   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))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 45.6

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

   10. Applied rewrites 45.6%

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

   12. Applied rewrites 45.8%

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

   if -9.99999999999999992e-221 < (/.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 18.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 6.6%

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

   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. 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. lower-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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B \cdot B\right)\right)}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B\_m \cdot B\_m\right)\right)}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) (- 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 (- INFINITY))
     (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) t_0)
     (if (<= t_2 -1e-220)
       (/ (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (- B_m))
       (if (<= t_2 INFINITY)
         (* t_0 (sqrt (/ (* -0.5 (* F (* B_m B_m))) A)))
         (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / -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 <= -((double) INFINITY)) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * t_0;
	} else if (t_2 <= -1e-220) {
		tmp = sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / -B_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0 * sqrt(((-0.5 * (F * (B_m * B_m))) / A));
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / Float64(-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 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * t_0);
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) / Float64(-B_m));
	elseif (t_2 <= Inf)
		tmp = Float64(t_0 * sqrt(Float64(Float64(-0.5 * Float64(F * Float64(B_m * B_m))) / A)));
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $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, (-Infinity)], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$0 * N[Sqrt[N[(N[(-0.5 * N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $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

   1. Initial program 3.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 8.4%

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

   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))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 45.6

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

   10. Applied rewrites 45.6%

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

   12. Applied rewrites 45.9%

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

   if -9.99999999999999992e-221 < (/.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 18.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 6.6%

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

   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. 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. lower-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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B \cdot B\right)\right)}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B\_m \cdot B\_m\right)\right)}{A}}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{-B\_m}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (* (/ (sqrt 2.0) (- B_m)) (sqrt (/ (* -0.5 (* F (* B_m B_m))) A))))
        (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 (- INFINITY))
     t_0
     (if (<= t_2 -1e-220)
       (/ (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (- B_m))
       (if (<= t_2 INFINITY) t_0 (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (sqrt(2.0) / -B_m) * sqrt(((-0.5 * (F * (B_m * B_m))) / A));
	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 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_2 <= -1e-220) {
		tmp = sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / -B_m;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = -sqrt(F) * sqrt((2.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(-0.5 * Float64(F * Float64(B_m * B_m))) / A)))
	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 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_2 <= -1e-220)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) / Float64(-B_m));
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $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, (-Infinity)], t$95$0, If[LessEqual[t$95$2, -1e-220], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$0, N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -9.99999999999999992e-221 < (/.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 12.2%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 4.9%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 6.4%

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

   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))) < -9.99999999999999992e-221

   1. Initial program 98.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 20.9

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

   5. Applied rewrites 20.9%

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

   7. Applied rewrites 20.8%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 45.6

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

   10. Applied rewrites 45.6%

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

   12. Applied rewrites 45.9%

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

   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. lower-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. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.6%

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

   9. Applied rewrites 29.6%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B \cdot B\right)\right)}{A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{-B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\frac{-0.5 \cdot \left(F \cdot \left(B \cdot B\right)\right)}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-120}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t\_0\right)} \cdot \frac{-\sqrt{F}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)}}{-B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-120)
     (* (sqrt (* 2.0 (* (* 2.0 C) t_0))) (/ (- (sqrt F)) t_0))
     (if (<= (pow B_m 2.0) 5e-14)
       (/
        (*
         (sqrt F)
         (sqrt
          (*
           (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))
           (* 2.0 (fma B_m B_m (* -4.0 (* A C)))))))
        (- (* B_m B_m)))
       (if (<= (pow B_m 2.0) 1e+167)
         (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
         (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-120) {
		tmp = sqrt((2.0 * ((2.0 * C) * t_0))) * (-sqrt(F) / t_0);
	} else if (pow(B_m, 2.0) <= 5e-14) {
		tmp = (sqrt(F) * sqrt((((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))) * (2.0 * fma(B_m, B_m, (-4.0 * (A * C))))))) / -(B_m * B_m);
	} else if (pow(B_m, 2.0) <= 1e+167) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

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(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-120)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * t_0))) * Float64(Float64(-sqrt(F)) / t_0));
	elseif ((B_m ^ 2.0) <= 5e-14)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * Float64(2.0 * fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))))) / Float64(-Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 1e+167)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-120], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-14], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+167], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999979e-121

   1. Initial program 23.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 13.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 15.1

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

   7. Applied rewrites 15.1%

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

   9. Applied rewrites 15.1%

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

   if 9.99999999999999979e-121 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e-14

   1. Initial program 48.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in B around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 52.5

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

   7. Applied rewrites 52.5%

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

   if 5.0000000000000002e-14 < (pow.f64 B #s(literal 2 binary64)) < 1e167

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 10.8%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.1%

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

   if 1e167 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.2%

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 32.2

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

   5. Applied rewrites 32.2%

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

   7. Applied rewrites 45.8%

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

   9. Applied rewrites 45.8%

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

   11. Applied rewrites 45.9%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-120}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}}{-B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := -\sqrt{F}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-120}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t\_0\right)} \cdot \frac{t\_1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(t\_1 \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B_m B_m))) (t_1 (- (sqrt F))))
   (if (<= (pow B_m 2.0) 1e-120)
     (* (sqrt (* 2.0 (* (* 2.0 C) t_0))) (/ t_1 t_0))
     (if (<= (pow B_m 2.0) 5e-31)
       (* (/ (sqrt 2.0) B_m) (* t_1 (sqrt (+ C (sqrt (fma B_m B_m (* C C)))))))
       (if (<= (pow B_m 2.0) 1e+167)
         (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
         (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B_m * B_m));
	double t_1 = -sqrt(F);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-120) {
		tmp = sqrt((2.0 * ((2.0 * C) * t_0))) * (t_1 / t_0);
	} else if (pow(B_m, 2.0) <= 5e-31) {
		tmp = (sqrt(2.0) / B_m) * (t_1 * sqrt((C + sqrt(fma(B_m, B_m, (C * C))))));
	} else if (pow(B_m, 2.0) <= 1e+167) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

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(C, Float64(A * -4.0), Float64(B_m * B_m))
	t_1 = Float64(-sqrt(F))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-120)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(2.0 * C) * t_0))) * Float64(t_1 / t_0));
	elseif ((B_m ^ 2.0) <= 5e-31)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(t_1 * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	elseif ((B_m ^ 2.0) <= 1e+167)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[F], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-120], N[(N[Sqrt[N[(2.0 * N[(N[(2.0 * C), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-31], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+167], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999979e-121

   1. Initial program 23.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 13.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 15.1

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

   7. Applied rewrites 15.1%

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

   9. Applied rewrites 15.1%

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

   if 9.99999999999999979e-121 < (pow.f64 B #s(literal 2 binary64)) < 5e-31

   1. Initial program 54.5%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 8.9%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 28.8

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

   10. Applied rewrites 28.8%

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

   12. Applied rewrites 35.2%

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

   if 5e-31 < (pow.f64 B #s(literal 2 binary64)) < 1e167

   1. Initial program 18.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.9%

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

   if 1e167 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.2%

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 32.2

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

   5. Applied rewrites 32.2%

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

   7. Applied rewrites 45.8%

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

   9. Applied rewrites 45.8%

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

   11. Applied rewrites 45.9%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-120}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-132}:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-132)
     (* (sqrt (* (* 2.0 C) (* 2.0 (* F t_0)))) (/ -1.0 t_0))
     (if (<= (pow B_m 2.0) 5e-31)
       (*
        (/ (sqrt 2.0) B_m)
        (* (- (sqrt F)) (sqrt (+ C (sqrt (fma B_m B_m (* C C)))))))
       (if (<= (pow B_m 2.0) 1e+167)
         (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
         (- (/ (sqrt F) (sqrt (* B_m 0.5)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-132) {
		tmp = sqrt(((2.0 * C) * (2.0 * (F * t_0)))) * (-1.0 / t_0);
	} else if (pow(B_m, 2.0) <= 5e-31) {
		tmp = (sqrt(2.0) / B_m) * (-sqrt(F) * sqrt((C + sqrt(fma(B_m, B_m, (C * C))))));
	} else if (pow(B_m, 2.0) <= 1e+167) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

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(C, Float64(A * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-132)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(F * t_0)))) * Float64(-1.0 / t_0));
	elseif ((B_m ^ 2.0) <= 5e-31)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(Float64(-sqrt(F)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	elseif ((B_m ^ 2.0) <= 1e+167)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-132], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-31], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+167], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999999e-133

   1. Initial program 22.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.5

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

   7. Applied rewrites 14.5%

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

   9. Applied rewrites 17.7%

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

   if 9.9999999999999999e-133 < (pow.f64 B #s(literal 2 binary64)) < 5e-31

   1. Initial program 48.3%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 8.1

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

   5. Applied rewrites 8.1%

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   12. Applied rewrites 28.1%

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

   if 5e-31 < (pow.f64 B #s(literal 2 binary64)) < 1e167

   1. Initial program 18.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 13.1%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 9.9%

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

   if 1e167 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.2%

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 32.2

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

   5. Applied rewrites 32.2%

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

   7. Applied rewrites 45.8%

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

   9. Applied rewrites 45.8%

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

   11. Applied rewrites 45.9%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-132}:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-132)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* F (fma C (* A -4.0) (* B_m B_m))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= (pow B_m 2.0) 5e-31)
     (*
      (/ (sqrt 2.0) B_m)
      (* (- (sqrt F)) (sqrt (+ C (sqrt (fma B_m B_m (* C C)))))))
     (if (<= (pow B_m 2.0) 1e+167)
       (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
       (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-132) {
		tmp = sqrt(((2.0 * C) * (2.0 * (F * fma(C, (A * -4.0), (B_m * B_m)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e-31) {
		tmp = (sqrt(2.0) / B_m) * (-sqrt(F) * sqrt((C + sqrt(fma(B_m, B_m, (C * C))))));
	} else if (pow(B_m, 2.0) <= 1e+167) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

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-132)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(F * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e-31)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(Float64(-sqrt(F)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))));
	elseif ((B_m ^ 2.0) <= 1e+167)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-132], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-31], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+167], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999999e-133

   1. Initial program 22.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

   4. Applied rewrites 13.2%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 14.5

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

   7. Applied rewrites 14.5%

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

   9. Applied rewrites 17.7%

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

   if 9.9999999999999999e-133 < (pow.f64 B #s(literal 2 binary64)) < 5e-31

   1. Initial program 48.3%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 8.1

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

   5. Applied rewrites 8.1%

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 23.1

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

   10. Applied rewrites 23.1%

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

   12. Applied rewrites 28.1%

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

   if 5e-31 < (pow.f64 B #s(literal 2 binary64)) < 1e167

   1. Initial program 18.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

   5. Applied rewrites 13.1%

      \[\leadsto \color{blue}{\sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} \cdot \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 9.9%

      \[\leadsto \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]

   if 1e167 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-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. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. lower-/.f64 32.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 32.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.8%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 50.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B\_m \cdot B\_m\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B\_m \cdot B\_m}{A}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-132)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* F (fma C (* A -4.0) (* B_m B_m))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (if (<= (pow B_m 2.0) 5e-31)
     (* (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (/ -1.0 B_m))
     (if (<= (pow B_m 2.0) 1e+167)
       (* (sqrt (* F (* -0.5 (/ (* B_m B_m) A)))) (/ (sqrt 2.0) (- B_m)))
       (- (/ (sqrt F) (sqrt (* B_m 0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-132) {
		tmp = sqrt(((2.0 * C) * (2.0 * (F * fma(C, (A * -4.0), (B_m * B_m)))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e-31) {
		tmp = sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))) * (-1.0 / B_m);
	} else if (pow(B_m, 2.0) <= 1e+167) {
		tmp = sqrt((F * (-0.5 * ((B_m * B_m) / A)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

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-132)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(F * fma(C, Float64(A * -4.0), Float64(B_m * B_m)))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e-31)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) * Float64(-1.0 / B_m));
	elseif ((B_m ^ 2.0) <= 1e+167)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64(Float64(B_m * B_m) / A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-132], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-31], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+167], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999999e-133

   1. Initial program 22.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\left(\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot F}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \cdot \sqrt{F}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. pow1/2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 13.2%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} + \left(A + C\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lower-*.f64 14.5

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 14.5%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]

   if 9.9999999999999999e-133 < (pow.f64 B #s(literal 2 binary64)) < 5e-31

   1. Initial program 48.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\color{blue}{\frac{-1}{2} \cdot {B}^{2}}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\frac{-1}{2} \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 8.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 8.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 12.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot 2} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, \left(-4 \cdot C\right) \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 23.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)} \]

   10. Applied rewrites 23.1%

       \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 23.4%

       \[\leadsto -\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)} \cdot \frac{1}{B} \]

   if 5e-31 < (pow.f64 B #s(literal 2 binary64)) < 1e167

   1. Initial program 18.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \frac{\sqrt{2}}{B}}\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]

   5. Applied rewrites 13.1%

      \[\leadsto \color{blue}{\sqrt{F \cdot \left(A + \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)} \]

   6. Taylor expanded in A around -inf

      \[\leadsto \sqrt{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} \cdot \left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 9.9%

      \[\leadsto \sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]

   if 1e167 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lower-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. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. lower-/.f64 32.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 32.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
   8. Step-by-step derivation

   9. Applied rewrites 45.8%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)} \cdot \frac{-1}{B}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+167}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+79)
   (* (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (/ -1.0 B_m))
   (- (/ (sqrt F) (sqrt (* B_m 0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+79) {
		tmp = sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))) * (-1.0 / B_m);
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+79)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+79], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e79

   1. Initial program 27.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\color{blue}{\frac{-1}{2} \cdot {B}^{2}}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\frac{-1}{2} \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 16.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 16.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 16.0%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot 2} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, \left(-4 \cdot C\right) \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 10.0

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)} \]

   10. Applied rewrites 10.0%

       \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 10.0%

       \[\leadsto -\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)} \cdot \frac{1}{B} \]

   if 5e79 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. lower-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. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. lower-/.f64 28.9

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 28.9%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.1%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
   8. Step-by-step derivation

   9. Applied rewrites 40.1%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 40.2%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+114}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+114)
   (/ (sqrt (* 2.0 (* F (+ C (sqrt (fma B_m B_m (* C C))))))) (- B_m))
   (- (/ (sqrt F) (sqrt (* B_m 0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+114) {
		tmp = sqrt((2.0 * (F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / -B_m;
	} else {
		tmp = -(sqrt(F) / sqrt((B_m * 0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+114)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))) / Float64(-B_m));
	else
		tmp = Float64(-Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+114], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e114

   1. Initial program 26.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{\frac{-1}{2} \cdot {B}^{2}}{A}}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\color{blue}{\frac{-1}{2} \cdot {B}^{2}}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{\frac{-1}{2} \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-*.f64 17.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 17.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 16.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot 2} \cdot \sqrt{F \cdot \mathsf{fma}\left(B, B, \left(-4 \cdot C\right) \cdot A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   9. 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. lower-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. lower-*.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. lower-/.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. lower-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. lower-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. lower-*.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. lower-+.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. lower-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. lower-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. lower-*.f64 10.2

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}\right)} \]

   10. Applied rewrites 10.2%

       \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 10.2%

       \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{\color{blue}{-B}} \]

   if 1e114 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. lower-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. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      7. lower-/.f64 30.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 30.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.3%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
   8. Step-by-step derivation

   9. Applied rewrites 42.3%

      \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 35.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(F) * sqrt((2.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) * sqrt((2.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) * Math.sqrt((2.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) * math.sqrt((2.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) * sqrt((2.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.4%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. lower-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. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 18.4

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 18.4%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 24.1%

   \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
8. Step-by-step derivation

9. Applied rewrites 24.1%

   \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]

10. Final simplification 24.1%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 22: 27.5% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.4%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in 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. lower-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. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   7. lower-/.f64 18.4

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 18.4%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 18.5%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

9. Applied rewrites 18.5%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))