ABCF->ab-angle a

Percentage Accurate: 19.5% → 63.5%

Time: 19.6s

Alternatives: 20

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: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_2 := -\sqrt{2 \cdot F}\\ t_3 := \frac{\sqrt{2 \cdot C}}{t\_0}\\ t_4 := \left(4 \cdot A\right) \cdot C\\ t_5 := t\_4 - {B\_m}^{2}\\ t_6 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_4\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_5}\\ \mathbf{if}\;t\_6 \leq -1 \cdot 10^{+203}:\\ \;\;\;\;t\_3 \cdot \left(\sqrt{t\_1} \cdot t\_2\right)\\ \mathbf{elif}\;t\_6 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right) \cdot \left(2 \cdot t\_1\right)} \cdot \sqrt{F}}{t\_5}\\ \mathbf{elif}\;t\_6 \leq \infty:\\ \;\;\;\;t\_3 \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (fma B_m B_m (* -4.0 (* A C))))
        (t_2 (- (sqrt (* 2.0 F))))
        (t_3 (/ (sqrt (* 2.0 C)) t_0))
        (t_4 (* (* 4.0 A) C))
        (t_5 (- t_4 (pow B_m 2.0)))
        (t_6
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_4) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_5)))
   (if (<= t_6 -1e+203)
     (* t_3 (* (sqrt t_1) t_2))
     (if (<= t_6 -1e-204)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        (- t_0))
       (if (<= t_6 0.0)
         (/
          (*
           (sqrt (* (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)) (* 2.0 t_1)))
           (sqrt F))
          t_5)
         (if (<= t_6 INFINITY)
           (* t_3 (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) t_2)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_2 = -sqrt((2.0 * F));
	double t_3 = sqrt((2.0 * C)) / t_0;
	double t_4 = (4.0 * A) * C;
	double t_5 = t_4 - pow(B_m, 2.0);
	double t_6 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_4) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_5;
	double tmp;
	if (t_6 <= -1e+203) {
		tmp = t_3 * (sqrt(t_1) * t_2);
	} else if (t_6 <= -1e-204) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / -t_0;
	} else if (t_6 <= 0.0) {
		tmp = (sqrt((fma(-0.5, ((B_m * B_m) / A), (2.0 * C)) * (2.0 * t_1))) * sqrt(F)) / t_5;
	} else if (t_6 <= ((double) INFINITY)) {
		tmp = t_3 * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-sqrt(Float64(2.0 * F)))
	t_3 = Float64(sqrt(Float64(2.0 * C)) / t_0)
	t_4 = Float64(Float64(4.0 * A) * C)
	t_5 = Float64(t_4 - (B_m ^ 2.0))
	t_6 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_4) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_5)
	tmp = 0.0
	if (t_6 <= -1e+203)
		tmp = Float64(t_3 * Float64(sqrt(t_1) * t_2));
	elseif (t_6 <= -1e-204)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / Float64(-t_0));
	elseif (t_6 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)) * Float64(2.0 * t_1))) * sqrt(F)) / t_5);
	elseif (t_6 <= Inf)
		tmp = Float64(t_3 * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$4), $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$5), $MachinePrecision]}, If[LessEqual[t$95$6, -1e+203], N[(t$95$3 * N[(N[Sqrt[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, -1e-204], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$6, Infinity], N[(t$95$3 * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$2), $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))) < -9.9999999999999999e202

   1. Initial program 4.5%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 28.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. pow1/2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 37.2%

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

   if -9.9999999999999999e202 < (/.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))) < -1e-204

   1. Initial program 96.5%

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

   4. Applied rewrites 96.3%

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

   6. Applied rewrites 96.5%

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

   if -1e-204 < (/.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 3.4%

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

   4. Applied rewrites 15.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 45.9

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

   7. Applied rewrites 45.9%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.7%

   \[\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 -1 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\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^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 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}}{\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 C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_2 := -\sqrt{2 \cdot F}\\ t_3 := \frac{\sqrt{2 \cdot C}}{t\_0}\\ t_4 := \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_4\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4 - {B\_m}^{2}}\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+203}:\\ \;\;\;\;t\_3 \cdot \left(\sqrt{t\_1} \cdot t\_2\right)\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-t\_0}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_1}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_3 \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m)))
        (t_1 (fma B_m B_m (* -4.0 (* A C))))
        (t_2 (- (sqrt (* 2.0 F))))
        (t_3 (/ (sqrt (* 2.0 C)) t_0))
        (t_4 (* (* 4.0 A) C))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_4) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_4 (pow B_m 2.0)))))
   (if (<= t_5 -1e+203)
     (* t_3 (* (sqrt t_1) t_2))
     (if (<= t_5 -1e-204)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
        (- t_0))
       (if (<= t_5 0.0)
         (*
          (/ (sqrt F) -1.0)
          (/ (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m)))))) t_1))
         (if (<= t_5 INFINITY)
           (* t_3 (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) t_2)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_2 = -sqrt((2.0 * F));
	double t_3 = sqrt((2.0 * C)) / t_0;
	double t_4 = (4.0 * A) * C;
	double t_5 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_4) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_4 - pow(B_m, 2.0));
	double tmp;
	if (t_5 <= -1e+203) {
		tmp = t_3 * (sqrt(t_1) * t_2);
	} else if (t_5 <= -1e-204) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / -t_0;
	} else if (t_5 <= 0.0) {
		tmp = (sqrt(F) / -1.0) * (sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) / t_1);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3 * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * t_2;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(-sqrt(Float64(2.0 * F)))
	t_3 = Float64(sqrt(Float64(2.0 * C)) / t_0)
	t_4 = Float64(Float64(4.0 * A) * C)
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_4) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_4 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_5 <= -1e+203)
		tmp = Float64(t_3 * Float64(sqrt(t_1) * t_2));
	elseif (t_5 <= -1e-204)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / Float64(-t_0));
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(sqrt(F) / -1.0) * Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) / t_1));
	elseif (t_5 <= Inf)
		tmp = Float64(t_3 * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * t_2);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$3 = N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$4), $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$4 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e+203], N[(t$95$3 * N[(N[Sqrt[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -1e-204], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(t$95$3 * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$2), $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))) < -9.9999999999999999e202

   1. Initial program 4.5%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.6

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

   5. Applied rewrites 22.6%

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

   7. Applied rewrites 28.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. pow1/2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 37.2%

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

   if -9.9999999999999999e202 < (/.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))) < -1e-204

   1. Initial program 96.5%

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

   4. Applied rewrites 96.3%

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

   6. Applied rewrites 96.5%

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

   if -1e-204 < (/.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 3.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 39.2%

   \[\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 -1 \cdot 10^{+203}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\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^{-204}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := -\sqrt{2 \cdot F}\\ t_2 := \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;t\_2 \cdot \left(\sqrt{t\_0} \cdot t\_1\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2 \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot t\_1\\ \end{array} \end{array} \]

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 (- (sqrt (* 2.0 F))))
        (t_2 (/ (sqrt (* 2.0 C)) (fma A (* C -4.0) (* B_m B_m))))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0)))))
   (if (<= t_4 -2e+162)
     (* t_2 (* (sqrt t_0) t_1))
     (if (<= t_4 -5e-153)
       (*
        (sqrt
         (/ (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_4 0.0)
         (*
          (/ (sqrt F) -1.0)
          (/ (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m)))))) t_0))
         (if (<= t_4 INFINITY)
           (* t_2 (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) t_1)))))))

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 = -sqrt((2.0 * F));
	double t_2 = sqrt((2.0 * C)) / fma(A, (C * -4.0), (B_m * B_m));
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -2e+162) {
		tmp = t_2 * (sqrt(t_0) * t_1);
	} else if (t_4 <= -5e-153) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_4 <= 0.0) {
		tmp = (sqrt(F) / -1.0) * (sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) / t_0);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2 * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * t_1;
	}
	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(-sqrt(Float64(2.0 * F)))
	t_2 = Float64(sqrt(Float64(2.0 * C)) / fma(A, Float64(C * -4.0), Float64(B_m * B_m)))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -2e+162)
		tmp = Float64(t_2 * Float64(sqrt(t_0) * t_1));
	elseif (t_4 <= -5e-153)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(sqrt(F) / -1.0) * Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) / t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(t_2 * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * t_1);
	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[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+162], N[(t$95$2 * N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-153], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$2 * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * t$95$1), $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))) < -1.9999999999999999e162

   1. Initial program 8.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 28.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. pow1/2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

   9. Applied rewrites 37.1%

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

   if -1.9999999999999999e162 < (/.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))) < -5.00000000000000033e-153

   1. Initial program 99.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if -5.00000000000000033e-153 < (/.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 8.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{2 \cdot F}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\mathsf{fma}\left(2, B\_m \cdot B\_m, \left(A \cdot C\right) \cdot -8\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (/ (sqrt (* 2.0 C)) (fma A (* C -4.0) (* B_m B_m))))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+162)
     (* t_1 (* (sqrt (fma 2.0 (* B_m B_m) (* (* A C) -8.0))) (- (sqrt F))))
     (if (<= t_3 -5e-153)
       (*
        (sqrt
         (/ (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_3 0.0)
         (*
          (/ (sqrt F) -1.0)
          (/ (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m)))))) t_0))
         (if (<= t_3 INFINITY)
           (* t_1 (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * C)) / fma(A, (C * -4.0), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+162) {
		tmp = t_1 * (sqrt(fma(2.0, (B_m * B_m), ((A * C) * -8.0))) * -sqrt(F));
	} else if (t_3 <= -5e-153) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) / -1.0) * (sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = Float64(sqrt(Float64(2.0 * C)) / fma(A, Float64(C * -4.0), Float64(B_m * B_m)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+162)
		tmp = Float64(t_1 * Float64(sqrt(fma(2.0, Float64(B_m * B_m), Float64(Float64(A * C) * -8.0))) * Float64(-sqrt(F))));
	elseif (t_3 <= -5e-153)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) / -1.0) * Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+162], N[(t$95$1 * N[(N[Sqrt[N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-153], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -1.9999999999999999e162

   1. Initial program 8.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 28.7%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. sqrt-prod N/A

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

      12. pow1/2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lift-*.f64 N/A

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

      18. unpow-prod-down N/A

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

      19. pow1/2 N/A

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

      20. lift-sqrt.f64 N/A

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

      21. associate-*r* N/A

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

   9. Applied rewrites 37.0%

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

   if -1.9999999999999999e162 < (/.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))) < -5.00000000000000033e-153

   1. Initial program 99.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if -5.00000000000000033e-153 < (/.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 8.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(2, B \cdot B, \left(A \cdot C\right) \cdot -8\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+162)
     (* t_1 (/ (sqrt (* t_0 (* 2.0 F))) (- t_0)))
     (if (<= t_3 -5e-153)
       (*
        (sqrt
         (/ (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_3 0.0)
         (*
          (/ (sqrt F) -1.0)
          (/ (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m)))))) t_0))
         (if (<= t_3 INFINITY)
           (*
            (/ t_1 (fma A (* C -4.0) (* B_m B_m)))
            (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+162) {
		tmp = t_1 * (sqrt((t_0 * (2.0 * F))) / -t_0);
	} else if (t_3 <= -5e-153) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) / -1.0) * (sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 / fma(A, (C * -4.0), (B_m * B_m))) * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+162)
		tmp = Float64(t_1 * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_3 <= -5e-153)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) / -1.0) * Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+162], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-153], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -1.9999999999999999e162

   1. Initial program 8.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 28.7%

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

   9. Applied rewrites 28.8%

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

   if -1.9999999999999999e162 < (/.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))) < -5.00000000000000033e-153

   1. Initial program 99.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if -5.00000000000000033e-153 < (/.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 8.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.7%

   \[\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 -2 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot C} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\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 -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F}}{-1} \cdot \frac{\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+162}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{t\_0}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)} \cdot \frac{-\sqrt{F}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+162)
     (* t_1 (/ (sqrt (* t_0 (* 2.0 F))) (- t_0)))
     (if (<= t_3 -5e-153)
       (*
        (sqrt
         (/ (* F (+ (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C)))))) t_0))
        (- (sqrt 2.0)))
       (if (<= t_3 0.0)
         (*
          (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m))))))
          (/ (- (sqrt F)) t_0))
         (if (<= t_3 INFINITY)
           (*
            (/ t_1 (fma A (* C -4.0) (* B_m B_m)))
            (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+162) {
		tmp = t_1 * (sqrt((t_0 * (2.0 * F))) / -t_0);
	} else if (t_3 <= -5e-153) {
		tmp = sqrt(((F * ((A + C) + sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / t_0)) * -sqrt(2.0);
	} else if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) * (-sqrt(F) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 / fma(A, (C * -4.0), (B_m * B_m))) * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+162)
		tmp = Float64(t_1 * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_3 <= -5e-153)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / t_0)) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) * Float64(Float64(-sqrt(F)) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+162], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-153], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -1.9999999999999999e162

   1. Initial program 8.7%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 23.2

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

   5. Applied rewrites 23.2%

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

   7. Applied rewrites 28.7%

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

   9. Applied rewrites 28.8%

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

   if -1.9999999999999999e162 < (/.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))) < -5.00000000000000033e-153

   1. Initial program 99.4%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

   if -5.00000000000000033e-153 < (/.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 8.2%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.4

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

   5. Applied rewrites 28.4%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.7%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.7%

   \[\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 -2 \cdot 10^{+162}:\\ \;\;\;\;\sqrt{2 \cdot C} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\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 -5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)} \cdot \frac{-\sqrt{F}}{\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{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \sqrt{2 \cdot F}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\left(t\_4 \cdot \left(-B\_m\right)\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)} \cdot \frac{-\sqrt{F}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-t\_4\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 C)))
        (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 (sqrt (* 2.0 F))))
   (if (<= t_3 -1e+154)
     (* t_1 (/ (sqrt (* t_0 (* 2.0 F))) (- t_0)))
     (if (<= t_3 -1e-204)
       (/ (* (* t_4 (- B_m)) (sqrt (+ C (sqrt (fma B_m B_m (* C C)))))) t_0)
       (if (<= t_3 0.0)
         (*
          (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m))))))
          (/ (- (sqrt F)) t_0))
         (if (<= t_3 INFINITY)
           (*
            (/ t_1 (fma A (* C -4.0) (* B_m B_m)))
            (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- t_4))))))))

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 = sqrt((2.0 * C));
	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 = sqrt((2.0 * F));
	double tmp;
	if (t_3 <= -1e+154) {
		tmp = t_1 * (sqrt((t_0 * (2.0 * F))) / -t_0);
	} else if (t_3 <= -1e-204) {
		tmp = ((t_4 * -B_m) * sqrt((C + sqrt(fma(B_m, B_m, (C * C)))))) / t_0;
	} else if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) * (-sqrt(F) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 / fma(A, (C * -4.0), (B_m * B_m))) * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -t_4;
	}
	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 = sqrt(Float64(2.0 * C))
	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 = sqrt(Float64(2.0 * F))
	tmp = 0.0
	if (t_3 <= -1e+154)
		tmp = Float64(t_1 * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_3 <= -1e-204)
		tmp = Float64(Float64(Float64(t_4 * Float64(-B_m)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) / t_0);
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) * Float64(Float64(-sqrt(F)) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-t_4));
	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[Sqrt[N[(2.0 * C), $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[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -1e+154], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-204], N[(N[(N[(t$95$4 * (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-t$95$4)), $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))) < -1.00000000000000004e154

   1. Initial program 10.0%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 28.3%

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

   9. Applied rewrites 28.4%

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

   if -1.00000000000000004e154 < (/.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))) < -1e-204

   1. Initial program 96.0%

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in A around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

   8. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-sqrt.f64 46.5

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

   10. Applied rewrites 46.5%

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

   12. Applied rewrites 46.5%

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

   if -1e-204 < (/.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 3.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot C} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;\frac{\left(\sqrt{2 \cdot F} \cdot \left(-B\right)\right) \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B\_m \cdot B\_m\right)\right)\right)} \cdot \frac{-\sqrt{F}}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -1e+154)
     (* t_1 (/ (sqrt (* t_0 (* 2.0 F))) (- t_0)))
     (if (<= t_3 -1e-204)
       (-
        (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) B_m))
       (if (<= t_3 0.0)
         (*
          (sqrt (* 2.0 (* C (fma -8.0 (* A C) (* 2.0 (* B_m B_m))))))
          (/ (- (sqrt F)) t_0))
         (if (<= t_3 INFINITY)
           (*
            (/ t_1 (fma A (* C -4.0) (* B_m B_m)))
            (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e+154) {
		tmp = t_1 * (sqrt((t_0 * (2.0 * F))) / -t_0);
	} else if (t_3 <= -1e-204) {
		tmp = -((sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * (C * fma(-8.0, (A * C), (2.0 * (B_m * B_m)))))) * (-sqrt(F) / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 / fma(A, (C * -4.0), (B_m * B_m))) * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e+154)
		tmp = Float64(t_1 * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_3 <= -1e-204)
		tmp = Float64(-Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C * fma(-8.0, Float64(A * C), Float64(2.0 * Float64(B_m * B_m)))))) * Float64(Float64(-sqrt(F)) / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+154], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-204], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * N[(C * N[(-8.0 * N[(A * C), $MachinePrecision] + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -1.00000000000000004e154

   1. Initial program 10.0%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 28.3%

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

   9. Applied rewrites 28.4%

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

   if -1.00000000000000004e154 < (/.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))) < -1e-204

   1. Initial program 96.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 46.3%

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

   if -1e-204 < (/.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 3.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 35.7%

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

   9. Applied rewrites 35.8%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot C} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot \left(C \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right)} \cdot \frac{-\sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_1 := \sqrt{2 \cdot C}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+154}:\\ \;\;\;\;t\_1 \cdot \frac{\sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B\_m, 2 \cdot B\_m, \left(A \cdot C\right) \cdot -8\right)}}{-4 \cdot \left(C \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* -4.0 (* A C))))
        (t_1 (sqrt (* 2.0 C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -1e+154)
     (* t_1 (/ (sqrt (* t_0 (* 2.0 F))) (- t_0)))
     (if (<= t_3 -1e-204)
       (-
        (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) B_m))
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt F)
           (sqrt (* (* 2.0 C) (fma B_m (* 2.0 B_m) (* (* A C) -8.0)))))
          (* -4.0 (* C (- A))))
         (if (<= t_3 INFINITY)
           (*
            (/ t_1 (fma A (* C -4.0) (* B_m B_m)))
            (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_1 = sqrt((2.0 * C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e+154) {
		tmp = t_1 * (sqrt((t_0 * (2.0 * F))) / -t_0);
	} else if (t_3 <= -1e-204) {
		tmp = -((sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((2.0 * C) * fma(B_m, (2.0 * B_m), ((A * C) * -8.0))))) / (-4.0 * (C * -A));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (t_1 / fma(A, (C * -4.0), (B_m * B_m))) * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_1 = sqrt(Float64(2.0 * C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e+154)
		tmp = Float64(t_1 * Float64(sqrt(Float64(t_0 * Float64(2.0 * F))) / Float64(-t_0)));
	elseif (t_3 <= -1e-204)
		tmp = Float64(-Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * C) * fma(B_m, Float64(2.0 * B_m), Float64(Float64(A * C) * -8.0))))) / Float64(-4.0 * Float64(C * Float64(-A))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(t_1 / fma(A, Float64(C * -4.0), Float64(B_m * B_m))) * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+154], N[(t$95$1 * N[(N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-204], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(B$95$m * N[(2.0 * B$95$m), $MachinePrecision] + N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(t$95$1 / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -1.00000000000000004e154

   1. Initial program 10.0%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.8

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

   5. Applied rewrites 22.8%

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

   7. Applied rewrites 28.3%

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

   9. Applied rewrites 28.4%

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

   if -1.00000000000000004e154 < (/.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))) < -1e-204

   1. Initial program 96.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 46.3%

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

   if -1e-204 < (/.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 3.4%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 35.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 35.5

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

   10. Applied rewrites 35.5%

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

   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 45.8%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 34.5

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

   5. Applied rewrites 34.5%

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

   7. Applied rewrites 40.7%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 34.8

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

   10. Applied rewrites 34.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*r* N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. sqrt-prod N/A

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

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

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

       16. lower-sqrt.f64 47.0

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

   12. Applied rewrites 47.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. 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 14.7

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

   5. Applied rewrites 14.7%

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

      1. lift-/.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. pow1/2 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-div N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 N/A

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

      23. metadata-eval N/A

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

      24. lower-sqrt.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. metadata-eval N/A

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

      2. sqrt-div N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 24.1

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

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2 \cdot C} \cdot \frac{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, 2 \cdot B, \left(A \cdot C\right) \cdot -8\right)}}{-4 \cdot \left(C \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(A \cdot C\right) \cdot -8\\ t_1 := \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+185}:\\ \;\;\;\;t\_1 \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{t\_0}\right)\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B\_m \cdot B\_m\right)}\right)}}{B\_m}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B\_m, 2 \cdot B\_m, t\_0\right)}}{-4 \cdot \left(C \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* A C) -8.0))
        (t_1 (/ (sqrt (* 2.0 C)) (fma A (* C -4.0) (* B_m B_m))))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -1e+185)
     (* t_1 (* (- (sqrt F)) (sqrt t_0)))
     (if (<= t_3 -1e-204)
       (-
        (/ (* (sqrt 2.0) (sqrt (* F (+ C (sqrt (fma C C (* B_m B_m))))))) B_m))
       (if (<= t_3 0.0)
         (/
          (* (sqrt F) (sqrt (* (* 2.0 C) (fma B_m (* 2.0 B_m) t_0))))
          (* -4.0 (* C (- A))))
         (if (<= t_3 INFINITY)
           (* t_1 (* (sqrt (* F (* A -8.0))) (- (sqrt C))))
           (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (A * C) * -8.0;
	double t_1 = sqrt((2.0 * C)) / fma(A, (C * -4.0), (B_m * B_m));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e+185) {
		tmp = t_1 * (-sqrt(F) * sqrt(t_0));
	} else if (t_3 <= -1e-204) {
		tmp = -((sqrt(2.0) * sqrt((F * (C + sqrt(fma(C, C, (B_m * B_m))))))) / B_m);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(F) * sqrt(((2.0 * C) * fma(B_m, (2.0 * B_m), t_0)))) / (-4.0 * (C * -A));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 * (sqrt((F * (A * -8.0))) * -sqrt(C));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(A * C) * -8.0)
	t_1 = Float64(sqrt(Float64(2.0 * C)) / fma(A, Float64(C * -4.0), Float64(B_m * B_m)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -1e+185)
		tmp = Float64(t_1 * Float64(Float64(-sqrt(F)) * sqrt(t_0)));
	elseif (t_3 <= -1e-204)
		tmp = Float64(-Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(C + sqrt(fma(C, C, Float64(B_m * B_m))))))) / B_m));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(2.0 * C) * fma(B_m, Float64(2.0 * B_m), t_0)))) / Float64(-4.0 * Float64(C * Float64(-A))));
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 * Float64(sqrt(Float64(F * Float64(A * -8.0))) * Float64(-sqrt(C))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -8.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+185], N[(t$95$1 * N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-204], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(C * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(B$95$m * N[(2.0 * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(t$95$1 * N[(N[Sqrt[N[(F * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $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))) < -9.9999999999999998e184

   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

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

      1. lower-*.f64 23.8

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

   5. Applied rewrites 23.8%

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.9

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

   10. Applied rewrites 21.9%

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

       1. associate-*r* N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. sqrt-prod N/A

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

       7. lift-sqrt.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-sqrt.f64 30.2

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 30.2

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 30.2

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

   12. Applied rewrites 30.2%

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

   if -9.9999999999999998e184 < (/.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))) < -1e-204

   1. Initial program 96.4%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{-B}} \]

   if -1e-204 < (/.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 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 28.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 35.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B \cdot 2, \left(A \cdot C\right) \cdot -8\right)} \cdot \sqrt{F}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Taylor expanded in B around 0

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B \cdot 2, \left(A \cdot C\right) \cdot -8\right)} \cdot \sqrt{F}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B \cdot 2, \left(A \cdot C\right) \cdot -8\right)} \cdot \sqrt{F}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

      2. lower-*.f64 35.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B \cdot 2, \left(A \cdot C\right) \cdot -8\right)} \cdot \sqrt{F}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   10. Applied rewrites 35.5%

       \[\leadsto \frac{-\sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, B \cdot 2, \left(A \cdot C\right) \cdot -8\right)} \cdot \sqrt{F}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   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 45.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 34.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 34.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}\right)} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      3. lower-*.f64 34.8

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)}\right) \]

   10. Applied rewrites 34.8%

       \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)}\right)\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot F\right)}}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(C \cdot F\right)}}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(F \cdot C\right)}}\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(\left(-8 \cdot A\right) \cdot F\right) \cdot C}}\right)\right) \]

       8. sqrt-prod N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \sqrt{C}}\right)\right) \]

       9. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot \color{blue}{{C}^{\frac{1}{2}}}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F} \cdot {C}^{\frac{1}{2}}}\right)\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\color{blue}{\sqrt{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot F}} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{\left(A \cdot -8\right)} \cdot F} \cdot {C}^{\frac{1}{2}}\right)\right) \]

       15. pow1/2 N/A

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\left(A \cdot -8\right) \cdot F} \cdot \color{blue}{\sqrt{C}}\right)\right) \]

       16. lower-sqrt.f64 47.0

           \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\left(A \cdot -8\right) \cdot F} \cdot \color{blue}{\sqrt{C}}\right) \]

   12. Applied rewrites 47.0%

       \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\color{blue}{\sqrt{\left(A \cdot -8\right) \cdot F} \cdot \sqrt{C}}\right) \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. 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 14.7

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.7%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 24.0

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 24.0%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 24.1

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 24.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{+185}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\left(-\sqrt{F}\right) \cdot \sqrt{\left(A \cdot C\right) \cdot -8}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-204}:\\ \;\;\;\;-\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{\left(2 \cdot C\right) \cdot \mathsf{fma}\left(B, 2 \cdot B, \left(A \cdot C\right) \cdot -8\right)}}{-4 \cdot \left(C \cdot \left(-A\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\sqrt{F \cdot \left(A \cdot -8\right)} \cdot \left(-\sqrt{C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+64)
     (* (sqrt (* C (* 4.0 (* F t_0)))) (/ -1.0 t_0))
     (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+64) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) * (-1.0 / t_0);
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+64)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+64], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e64

   1. Initial program 22.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 28.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{C \cdot \left(\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4\right)}\right)} \]

   if 5e64 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 22.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 30.1

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 30.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 30.1

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 30.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+64)
     (/ (sqrt (* C (* 4.0 (* F t_0)))) (- t_0))
     (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+64) {
		tmp = sqrt((C * (4.0 * (F * t_0)))) / -t_0;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+64)
		tmp = Float64(sqrt(Float64(C * Float64(4.0 * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+64], N[(N[Sqrt[N[(C * N[(4.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5e64

   1. Initial program 22.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 28.1

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.1%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot F\right) \cdot 4\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 5e64 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.2

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 22.2%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 30.1

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 30.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 30.1

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 30.1%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(4 \cdot \left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 0.02:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 0.02)
   (*
    (sqrt (* (* 2.0 C) (* A (* -8.0 (* C F)))))
    (/ -1.0 (fma B_m B_m (* -4.0 (* A C)))))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 0.02) {
		tmp = sqrt(((2.0 * C) * (A * (-8.0 * (C * F))))) * (-1.0 / fma(B_m, B_m, (-4.0 * (A * C))));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.02)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(A * Float64(-8.0 * Float64(C * F))))) * Float64(-1.0 / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.02], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(A * N[(-8.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 0.0200000000000000004

   1. Initial program 20.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 29.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 29.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}\right)} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      3. lower-*.f64 18.6

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)}\right) \]

   10. Applied rewrites 18.6%

       \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right) \]

   12. Applied rewrites 24.7%

       \[\leadsto \color{blue}{\left(-\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot -8\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \]

   if 0.0200000000000000004 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 22.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 22.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 29.0

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 29.0%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 29.0

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 29.0%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0.02:\\ \;\;\;\;\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 0.02:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 0.02)
   (/
    (sqrt (* (* 2.0 C) (* A (* -8.0 (* C F)))))
    (- (fma B_m B_m (* -4.0 (* A C)))))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 0.02) {
		tmp = sqrt(((2.0 * C) * (A * (-8.0 * (C * F))))) / -fma(B_m, B_m, (-4.0 * (A * C)));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.02)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(A * Float64(-8.0 * Float64(C * F))))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.02], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(A * N[(-8.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 0.0200000000000000004

   1. Initial program 20.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 29.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 29.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}\right)} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\mathsf{neg}\left(\sqrt{-8 \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}}\right)\right) \]

      3. lower-*.f64 18.6

         \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \color{blue}{\left(C \cdot F\right)}\right)}\right) \]

   10. Applied rewrites 18.6%

       \[\leadsto \frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}\right) \]

   12. Applied rewrites 24.7%

       \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot -8\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \]

   if 0.0200000000000000004 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 22.1

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 22.1%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 29.0

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 29.0%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 29.0

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 29.0%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0.02:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(A \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2.95 \cdot 10^{+33}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2.95e+33)
   (* -2.0 (sqrt (/ (* C F) (fma B_m B_m (* -4.0 (* A C))))))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2.95e+33) {
		tmp = -2.0 * sqrt(((C * F) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2.95e+33)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2.95e+33], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.95000000000000004e33

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 28.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{C \cdot F}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}} \]

      6. unpow2 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      7. metadata-eval N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

      9. lower-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}} \]

      10. lower-*.f64 17.7

          \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}} \]

   8. Applied rewrites 17.7%

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

   if 2.95000000000000004e33 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 22.0

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 22.0%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

      5. div-inv N/A

         \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      9. unpow-prod-down N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

      16. pow1/2 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

      18. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

      22. lower-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

      23. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

      24. lower-sqrt.f64 29.1

          \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

   7. Applied rewrites 29.1%

      \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

      2. sqrt-div N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

      4. lower-/.f64 29.2

         \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

   9. Applied rewrites 29.2%

      \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2.95 \cdot 10^{+33}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right) \end{array} \]

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 (/ 1.0 B_m)) (- (sqrt (* 2.0 F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
}

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((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
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((1.0 / B_m)) * -Math.sqrt((2.0 * F));
}

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((1.0 / B_m)) * -math.sqrt((2.0 * F))

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(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))))
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((1.0 / B_m)) * -sqrt((2.0 * F));
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[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 14.0

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   16. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

   18. sqrt-div N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

   22. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   24. lower-sqrt.f64 18.1

       \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

7. Applied rewrites 18.1%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{B}}\right) \]

   2. sqrt-div N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

   4. lower-/.f64 18.1

      \[\leadsto -\sqrt{2 \cdot F} \cdot \sqrt{\color{blue}{\frac{1}{B}}} \]

9. Applied rewrites 18.1%

   \[\leadsto -\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}} \]

10. Final simplification 18.1%

    \[\leadsto \sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right) \]
11. Add Preprocessing

Alternative 17: 28.7% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 1.65e+245)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* (sqrt (* C F)) (/ 2.0 (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.65e+245) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = sqrt((C * F)) * (2.0 / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 1.65d+245) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = sqrt((c * f)) * (2.0d0 / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 1.65e+245) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = Math.sqrt((C * F)) * (2.0 / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 1.65e+245:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = math.sqrt((C * F)) * (2.0 / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 1.65e+245)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(sqrt(Float64(C * F)) * Float64(2.0 / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 1.65e+245)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = sqrt((C * F)) * (2.0 / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 1.65e+245], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(2.0 / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.65000000000000005e245

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. 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 14.6

         \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Applied rewrites 14.6%

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      5. lift-neg.f64 14.6

         \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      15. lower-/.f64 14.7

          \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{2 \cdot F}}{B}}\right) \]

      18. lower-*.f64 14.7

          \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

   7. Applied rewrites 14.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]

   if 1.65000000000000005e245 < C

   1. Initial program 0.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. lower-*.f64 0.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 0.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 14.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot C}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}\right)} \]

   8. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2}}{B}} \cdot \sqrt{C \cdot F}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{B} \cdot \sqrt{C \cdot F}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{2}}{B} \cdot \sqrt{C \cdot F}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{2}{B} \cdot \color{blue}{\sqrt{C \cdot F}}\right) \]

      8. lower-*.f64 9.1

         \[\leadsto -\frac{2}{B} \cdot \sqrt{\color{blue}{C \cdot F}} \]

   10. Applied rewrites 9.1%

       \[\leadsto \color{blue}{-\frac{2}{B} \cdot \sqrt{C \cdot F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{2}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.1% 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 15.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 14.0

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   2. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   3. pow1/2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\frac{1}{2}}}\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\frac{F}{B}}\right)}^{\frac{1}{2}}\right) \]

   5. div-inv N/A

      \[\leadsto \mathsf{neg}\left({\left(2 \cdot \color{blue}{\left(F \cdot \frac{1}{B}\right)}\right)}^{\frac{1}{2}}\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(2 \cdot F\right) \cdot \frac{1}{B}\right)}}^{\frac{1}{2}}\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(\color{blue}{\left(F \cdot 2\right)} \cdot \frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   9. unpow-prod-down N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(F \cdot 2\right)}^{\frac{1}{2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}}\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot 2}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot {\left(\frac{1}{B}\right)}^{\frac{1}{2}}\right) \]

   16. pow1/2 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\sqrt{\frac{1}{B}}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \sqrt{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{B}}\right) \]

   18. sqrt-div N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}}\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\sqrt{\color{blue}{1}}}{\sqrt{B}}\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\sqrt{B}}\right) \]

   22. lower-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\sqrt{B}}}\right) \]

   23. metadata-eval N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{\color{blue}{1}}{\sqrt{B}}\right) \]

   24. lower-sqrt.f64 18.1

       \[\leadsto -\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}} \]

7. Applied rewrites 18.1%

   \[\leadsto -\color{blue}{\sqrt{2 \cdot F} \cdot \frac{1}{\sqrt{B}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot F}} \cdot \frac{1}{\sqrt{B}}\right) \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot F}} \cdot \frac{1}{\sqrt{B}}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot F} \cdot \frac{1}{\color{blue}{\sqrt{B}}}\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2 \cdot F}}{\sqrt{B}}}\right) \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2 \cdot F}}}{\sqrt{B}}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2 \cdot F}}{\color{blue}{\sqrt{B}}}\right) \]

   7. sqrt-div N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{2 \cdot F}{B}}}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{2 \cdot F}}{B}}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   10. associate-/l* N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   11. sqrt-prod N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \sqrt{\frac{2}{B}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

   14. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

   15. lower-/.f64 18.1

       \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

9. Applied rewrites 18.1%

   \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

10. Final simplification 18.1%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 19: 27.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (* 2.0 F) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(((2.0 * F) / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * f) / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(((2.0 * F) / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(((2.0 * F) / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(((2.0 * F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 15.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 14.0

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lift-neg.f64 14.0

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   9. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   15. lower-/.f64 14.0

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{2 \cdot F}}{B}}\right) \]

   18. lower-*.f64 14.0

       \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Add Preprocessing

Alternative 20: 27.8% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 15.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
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 14.0

      \[\leadsto -\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}}\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   5. lift-neg.f64 14.0

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}}\right) \]

   9. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \color{blue}{\frac{F}{B}}}\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   15. lower-/.f64 14.0

       \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   16. lift-*.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{2 \cdot F}}{B}}\right) \]

   18. lower-*.f64 14.0

       \[\leadsto -\sqrt{\frac{\color{blue}{2 \cdot F}}{B}} \]

7. Applied rewrites 14.0%

   \[\leadsto \color{blue}{-\sqrt{\frac{2 \cdot F}{B}}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   4. lower-/.f64 14.0

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied rewrites 14.0%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))