ABCF->ab-angle b

Percentage Accurate: 18.9% → 51.4%

Time: 20.9s

Alternatives: 23

Speedup: 6.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot t\_0\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B\_m \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_3)
     (if (<= t_4 -1e-202)
       (*
        (*
         (sqrt (* (* F t_0) -2.0))
         (sqrt (- (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ A C))))
        (/ -1.0 (fma B_m B_m (* C (* A -4.0)))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ A (fma (/ (* B_m B_m) C) -0.5 A)))) t_3)
         (* (/ -2.0 (* B_m (sqrt 2.0))) (sqrt (* F (- A (hypot B_m A))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) / t_3;
	} else if (t_4 <= -1e-202) {
		tmp = (sqrt(((F * t_0) * -2.0)) * sqrt((sqrt(fma((A - C), (A - C), (B_m * B_m))) - (A + C)))) * (-1.0 / fma(B_m, B_m, (C * (A * -4.0))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (A + fma(((B_m * B_m) / C), -0.5, A)))) / t_3;
	} else {
		tmp = (-2.0 / (B_m * sqrt(2.0))) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	elseif (t_4 <= -1e-202)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * t_0) * -2.0)) * sqrt(Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) - Float64(A + C)))) * Float64(-1.0 / fma(B_m, B_m, Float64(C * Float64(A * -4.0)))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + fma(Float64(Float64(B_m * B_m) / C), -0.5, A)))) / t_3);
	else
		tmp = Float64(Float64(-2.0 / Float64(B_m * sqrt(2.0))) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-202], N[(N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(-2.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

   7. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 98.4%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 28.5

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

   5. Applied rewrites 28.5%

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

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

   1. Initial program 0.0%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 22.4%

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

   11. Applied rewrites 22.4%

       \[\leadsto \frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{\color{blue}{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot t\_0\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B\_m \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2)))
   (if (<= t_3 (- INFINITY))
     (/ (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_2)
     (if (<= t_3 -1e-202)
       (*
        (*
         (sqrt (* (* F t_0) -2.0))
         (sqrt (- (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ A C))))
        (/ -1.0 (fma B_m B_m (* C (* A -4.0)))))
       (if (<= t_3 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (* (/ -2.0 (* B_m (sqrt 2.0))) (sqrt (* F (- A (hypot B_m A))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) / t_2;
	} else if (t_3 <= -1e-202) {
		tmp = (sqrt(((F * t_0) * -2.0)) * sqrt((sqrt(fma((A - C), (A - C), (B_m * B_m))) - (A + C)))) * (-1.0 / fma(B_m, B_m, (C * (A * -4.0))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 / (B_m * sqrt(2.0))) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) / t_2);
	elseif (t_3 <= -1e-202)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * t_0) * -2.0)) * sqrt(Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) - Float64(A + C)))) * Float64(-1.0 / fma(B_m, B_m, Float64(C * Float64(A * -4.0)))));
	elseif (t_3 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 / Float64(B_m * sqrt(2.0))) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -1e-202], N[(N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

   7. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 98.4%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 22.4%

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

   11. Applied rewrites 22.4%

       \[\leadsto \frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{\color{blue}{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot t\_0\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B\_m \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 (- INFINITY))
     (/ (* (sqrt t_0) (sqrt (* (+ A A) (* 2.0 F)))) (- t_0))
     (if (<= t_2 -1e-202)
       (*
        (*
         (sqrt (* (* F t_0) -2.0))
         (sqrt (- (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ A C))))
        (/ -1.0 (fma B_m B_m (* C (* A -4.0)))))
       (if (<= t_2 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (* (/ -2.0 (* B_m (sqrt 2.0))) (sqrt (* F (- A (hypot B_m A))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * sqrt(((A + A) * (2.0 * F)))) / -t_0;
	} else if (t_2 <= -1e-202) {
		tmp = (sqrt(((F * t_0) * -2.0)) * sqrt((sqrt(fma((A - C), (A - C), (B_m * B_m))) - (A + C)))) * (-1.0 / fma(B_m, B_m, (C * (A * -4.0))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 / (B_m * sqrt(2.0))) * sqrt((F * (A - hypot(B_m, A))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (t_2 <= -1e-202)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * t_0) * -2.0)) * sqrt(Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) - Float64(A + C)))) * Float64(-1.0 / fma(B_m, B_m, Float64(C * Float64(A * -4.0)))));
	elseif (t_2 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 / Float64(B_m * sqrt(2.0))) * sqrt(Float64(F * Float64(A - hypot(B_m, A)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], N[(N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

   7. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 98.4%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 22.4%

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

   11. Applied rewrites 22.4%

       \[\leadsto \frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{\color{blue}{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot t\_0\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 (- INFINITY))
     (/ (* (sqrt t_0) (sqrt (* (+ A A) (* 2.0 F)))) (- t_0))
     (if (<= t_2 -1e-202)
       (*
        (*
         (sqrt (* (* F t_0) -2.0))
         (sqrt (- (sqrt (fma (- A C) (- A C) (* B_m B_m))) (+ A C))))
        (/ -1.0 (fma B_m B_m (* C (* A -4.0)))))
       (if (<= t_2 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * sqrt(((A + A) * (2.0 * F)))) / -t_0;
	} else if (t_2 <= -1e-202) {
		tmp = (sqrt(((F * t_0) * -2.0)) * sqrt((sqrt(fma((A - C), (A - C), (B_m * B_m))) - (A + C)))) * (-1.0 / fma(B_m, B_m, (C * (A * -4.0))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (t_2 <= -1e-202)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * t_0) * -2.0)) * sqrt(Float64(sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))) - Float64(A + C)))) * Float64(-1.0 / fma(B_m, B_m, Float64(C * Float64(A * -4.0)))));
	elseif (t_2 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], N[(N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

   7. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 98.4%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot -2} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)} - \left(A + C\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{t\_1 \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(t\_1 \cdot \sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot \left(2 \cdot F\right)}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (sqrt t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (/ (* t_1 (sqrt (* (+ A A) (* 2.0 F)))) (- t_0))
     (if (<= t_3 -1e-202)
       (*
        (/ -1.0 (fma B_m B_m (* C (* A -4.0))))
        (*
         t_1
         (sqrt
          (* (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))) (* 2.0 F)))))
       (if (<= t_3 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = sqrt(t_0);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (t_1 * sqrt(((A + A) * (2.0 * F)))) / -t_0;
	} else if (t_3 <= -1e-202) {
		tmp = (-1.0 / fma(B_m, B_m, (C * (A * -4.0)))) * (t_1 * sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))) * (2.0 * F))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(t_1 * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (t_3 <= -1e-202)
		tmp = Float64(Float64(-1.0 / fma(B_m, B_m, Float64(C * Float64(A * -4.0)))) * Float64(t_1 * sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * Float64(2.0 * F)))));
	elseif (t_3 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(t$95$1 * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, -1e-202], N[(N[(-1.0 / N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

   7. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.7%

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

   6. Applied rewrites 96.9%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(2 \cdot F\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right) \cdot \left(\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 (- INFINITY))
     (/ (* (sqrt t_0) (sqrt (* (+ A A) (* 2.0 F)))) (- t_0))
     (if (<= t_2 -1e-202)
       (/
        (sqrt
         (*
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))
          (* (fma B_m B_m (* C (* A -4.0))) (* 2.0 F))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_2 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * sqrt(((A + A) * (2.0 * F)))) / -t_0;
	} else if (t_2 <= -1e-202) {
		tmp = sqrt((((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))) * (fma(B_m, B_m, (C * (A * -4.0))) * (2.0 * F)))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (t_2 <= -1e-202)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) * Float64(fma(B_m, B_m, Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_2 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 11.5

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

   5. Applied rewrites 11.5%

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

   7. Applied rewrites 11.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 26.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-202

   1. Initial program 97.0%

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

   4. Applied rewrites 95.8%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B\_m \cdot B\_m}{C} \cdot -0.5\right)\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 -2e+33)
     (/ (* (sqrt t_0) (sqrt (* (+ A A) (* 2.0 F)))) (- t_0))
     (if (<= t_2 -1e-202)
       (*
        (sqrt (* F (- A (sqrt (fma B_m B_m (* A A))))))
        (- (/ (sqrt 2.0) B_m)))
       (if (<= t_2 INFINITY)
         (-
          (/
           (sqrt (* (+ A (+ A (* (/ (* B_m B_m) C) -0.5))) (* t_0 (* 2.0 F))))
           t_0))
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -2e+33) {
		tmp = (sqrt(t_0) * sqrt(((A + A) * (2.0 * F)))) / -t_0;
	} else if (t_2 <= -1e-202) {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * A)))))) * -(sqrt(2.0) / B_m);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -(sqrt(((A + (A + (((B_m * B_m) / C) * -0.5))) * (t_0 * (2.0 * F)))) / t_0);
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -2e+33)
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / Float64(-t_0));
	elseif (t_2 <= -1e-202)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / B_m)));
	elseif (t_2 <= Inf)
		tmp = Float64(-Float64(sqrt(Float64(Float64(A + Float64(A + Float64(Float64(Float64(B_m * B_m) / C) * -0.5))) * Float64(t_0 * Float64(2.0 * F)))) / t_0));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+33], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$2, -1e-202], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], (-N[(N[Sqrt[N[(N[(A + N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e33

   1. Initial program 26.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 30.8%

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

   if -1.9999999999999999e33 < (/.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-202

   1. Initial program 97.2%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 28.5

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

   10. Applied rewrites 28.5%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;-\frac{\sqrt{\left(A + \left(A + \frac{B \cdot B}{C} \cdot -0.5\right)\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (- t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+33)
     (/ (* (sqrt t_0) (sqrt (* (+ A A) (* 2.0 F)))) t_1)
     (if (<= t_3 -1e-202)
       (*
        (sqrt (* F (- A (sqrt (fma B_m B_m (* A A))))))
        (- (/ (sqrt 2.0) B_m)))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* (+ A A) (* t_0 (* 2.0 F)))) t_1)
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+33) {
		tmp = (sqrt(t_0) * sqrt(((A + A) * (2.0 * F)))) / t_1;
	} else if (t_3 <= -1e-202) {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * A)))))) * -(sqrt(2.0) / B_m);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((A + A) * (t_0 * (2.0 * F)))) / t_1;
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+33)
		tmp = Float64(Float64(sqrt(t_0) * sqrt(Float64(Float64(A + A) * Float64(2.0 * F)))) / t_1);
	elseif (t_3 <= -1e-202)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / B_m)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(2.0 * F)))) / t_1);
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+33], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-202], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e33

   1. Initial program 26.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. sqrt-prod N/A

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

   9. Applied rewrites 30.8%

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

   if -1.9999999999999999e33 < (/.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-202

   1. Initial program 97.2%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(A + A\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, B\_m \cdot B\_m, \left(A \cdot C\right) \cdot -8\right)} \cdot \frac{\sqrt{F \cdot \left(A + A\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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)))
        (t_1 (- t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 -2e+33)
     (*
      (sqrt (fma 2.0 (* B_m B_m) (* (* A C) -8.0)))
      (/ (sqrt (* F (+ A A))) t_1))
     (if (<= t_3 -1e-202)
       (*
        (sqrt (* F (- A (sqrt (fma B_m B_m (* A A))))))
        (- (/ (sqrt 2.0) B_m)))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* (+ A A) (* t_0 (* 2.0 F)))) t_1)
         (*
          (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
          (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = -t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e+33) {
		tmp = sqrt(fma(2.0, (B_m * B_m), ((A * C) * -8.0))) * (sqrt((F * (A + A))) / t_1);
	} else if (t_3 <= -1e-202) {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * A)))))) * -(sqrt(2.0) / B_m);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((A + A) * (t_0 * (2.0 * F)))) / t_1;
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= -2e+33)
		tmp = Float64(sqrt(fma(2.0, Float64(B_m * B_m), Float64(Float64(A * C) * -8.0))) * Float64(sqrt(Float64(F * Float64(A + A))) / t_1));
	elseif (t_3 <= -1e-202)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / B_m)));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(2.0 * F)))) / t_1);
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+33], 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[(N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-202], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999999e33

   1. Initial program 26.7%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 19.3

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

   5. Applied rewrites 19.3%

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

   7. Applied rewrites 19.3%

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

   9. Applied rewrites 27.9%

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

   if -1.9999999999999999e33 < (/.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-202

   1. Initial program 97.2%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 36.6

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

   8. Applied rewrites 36.6%

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

   if -1e-202 < (/.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 17.9%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 28.3

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

   5. Applied rewrites 28.3%

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

   7. Applied rewrites 28.3%

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

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

   1. 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. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around 0

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

   12. Applied rewrites 16.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, B \cdot B, \left(A \cdot C\right) \cdot -8\right)} \cdot \frac{\sqrt{F \cdot \left(A + A\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := B\_m \cdot \sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{t\_1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{A \cdot F}\right)\\ \end{array} \end{array} \]

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))) (t_1 (* B_m (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 2e-110)
     (/ (sqrt (* (+ A A) (* t_0 (* 2.0 F)))) (- t_0))
     (if (<= (pow B_m 2.0) 4e+256)
       (/ (* -2.0 (sqrt (* F (- A (sqrt (fma A A (* B_m B_m))))))) t_1)
       (* (* -2.0 (/ 1.0 t_1)) (* (sqrt 2.0) (sqrt (* A F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double t_1 = B_m * sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-110) {
		tmp = sqrt(((A + A) * (t_0 * (2.0 * F)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 4e+256) {
		tmp = (-2.0 * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))))) / t_1;
	} else {
		tmp = (-2.0 * (1.0 / t_1)) * (sqrt(2.0) * sqrt((A * F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	t_1 = Float64(B_m * sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-110)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 4e+256)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m))))))) / t_1);
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / t_1)) * Float64(sqrt(2.0) * sqrt(Float64(A * F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-110], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+256], N[(N[(-2.0 * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-110

   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 C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.8

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

   5. Applied rewrites 21.8%

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

   7. Applied rewrites 21.8%

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

   if 2.0000000000000001e-110 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000001e256

   1. Initial program 42.1%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 39.9%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 21.8

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

   7. Applied rewrites 21.8%

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

   9. Applied rewrites 21.9%

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

   if 4.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 3.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \left(-\sqrt{A \cdot F} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{A \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := B\_m \cdot \sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{t\_0}\right) \cdot \left(\sqrt{2} \cdot \sqrt{A \cdot F}\right)\\ \end{array} \end{array} \]

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 (* B_m (sqrt 2.0))))
   (if (<= (pow B_m 2.0) 4e-114)
     (/
      (sqrt (* (+ A A) (* (fma A (* C -4.0) (* B_m B_m)) (* 2.0 F))))
      (* 4.0 (* A C)))
     (if (<= (pow B_m 2.0) 4e+256)
       (/ (* -2.0 (sqrt (* F (- A (sqrt (fma A A (* B_m B_m))))))) t_0)
       (* (* -2.0 (/ 1.0 t_0)) (* (sqrt 2.0) (sqrt (* A 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 = B_m * sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 4e-114) {
		tmp = sqrt(((A + A) * (fma(A, (C * -4.0), (B_m * B_m)) * (2.0 * F)))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 4e+256) {
		tmp = (-2.0 * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))))) / t_0;
	} else {
		tmp = (-2.0 * (1.0 / t_0)) * (sqrt(2.0) * sqrt((A * 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(B_m * sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-114)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(fma(A, Float64(C * -4.0), Float64(B_m * B_m)) * Float64(2.0 * F)))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 4e+256)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m))))))) / t_0);
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / t_0)) * Float64(sqrt(2.0) * sqrt(Float64(A * 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 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-114], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+256], N[(N[(-2.0 * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e-114

   1. Initial program 19.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.4

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

   5. Applied rewrites 20.4%

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

   7. Applied rewrites 20.4%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 20.3

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

   10. Applied rewrites 20.3%

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

   if 4.0000000000000002e-114 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000001e256

   1. Initial program 42.4%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 21.3

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 21.3%

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

   if 4.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 3.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \left(-\sqrt{A \cdot F} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{A \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{B\_m \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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) 4e-114)
   (/
    (sqrt (* (+ A A) (* (fma A (* C -4.0) (* B_m B_m)) (* 2.0 F))))
    (* 4.0 (* A C)))
   (if (<= (pow B_m 2.0) 4e+256)
     (/
      (* -2.0 (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))))
      (* B_m (sqrt 2.0)))
     (* 2.0 (* (sqrt (* A F)) (/ -1.0 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 (pow(B_m, 2.0) <= 4e-114) {
		tmp = sqrt(((A + A) * (fma(A, (C * -4.0), (B_m * B_m)) * (2.0 * F)))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 4e+256) {
		tmp = (-2.0 * sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m))))))) / (B_m * sqrt(2.0));
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / 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 ((B_m ^ 2.0) <= 4e-114)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(fma(A, Float64(C * -4.0), Float64(B_m * B_m)) * Float64(2.0 * F)))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 4e+256)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m))))))) / Float64(B_m * sqrt(2.0)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-114], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+256], N[(N[(-2.0 * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000002e-114

   1. Initial program 19.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 20.4

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

   5. Applied rewrites 20.4%

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

   7. Applied rewrites 20.4%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 20.3

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

   10. Applied rewrites 20.3%

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

   if 4.0000000000000002e-114 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000001e256

   1. Initial program 42.4%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 40.2%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 21.3

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 21.3%

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

   if 4.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 1.8

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

   7. Applied rewrites 1.8%

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

   9. Applied rewrites 1.8%

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

   10. Taylor expanded in A around -inf

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

   12. Applied rewrites 3.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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) 2e-110)
   (/
    (sqrt (* (+ A A) (* (fma A (* C -4.0) (* B_m B_m)) (* 2.0 F))))
    (* 4.0 (* A C)))
   (if (<= (pow B_m 2.0) 4e+256)
     (* (sqrt (* F (- A (sqrt (fma B_m B_m (* A A)))))) (- (/ (sqrt 2.0) B_m)))
     (* 2.0 (* (sqrt (* A F)) (/ -1.0 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 (pow(B_m, 2.0) <= 2e-110) {
		tmp = sqrt(((A + A) * (fma(A, (C * -4.0), (B_m * B_m)) * (2.0 * F)))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 4e+256) {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * A)))))) * -(sqrt(2.0) / B_m);
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / 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 ((B_m ^ 2.0) <= 2e-110)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(fma(A, Float64(C * -4.0), Float64(B_m * B_m)) * Float64(2.0 * F)))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 4e+256)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-110], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+256], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-110

   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 C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.8

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

   5. Applied rewrites 21.8%

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

   7. Applied rewrites 21.8%

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

   8. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 20.8

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

   10. Applied rewrites 20.8%

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

   if 2.0000000000000001e-110 < (pow.f64 B #s(literal 2 binary64)) < 4.0000000000000001e256

   1. Initial program 42.1%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 10.6

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

   5. Applied rewrites 10.6%

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

   6. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 21.9

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

   8. Applied rewrites 21.9%

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

   if 4.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 1.8

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 1.8%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 1.8%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 3.8%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right) \cdot \left(2 \cdot F\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot t\_0}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(-B\_m\right)}}{B\_m \cdot \left(-B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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)) (* 2.0 F))))
   (if (<= (pow B_m 2.0) 2e-101)
     (/ (sqrt (* (+ A A) t_0)) (* 4.0 (* A C)))
     (if (<= (pow B_m 2.0) 2e+256)
       (/ (sqrt (* t_0 (- B_m))) (* B_m (- B_m)))
       (* 2.0 (* (sqrt (* A F)) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m)) * (2.0 * F);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-101) {
		tmp = sqrt(((A + A) * t_0)) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 2e+256) {
		tmp = sqrt((t_0 * -B_m)) / (B_m * -B_m);
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(fma(A, Float64(C * -4.0), Float64(B_m * B_m)) * Float64(2.0 * F))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-101)
		tmp = Float64(sqrt(Float64(Float64(A + A) * t_0)) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 2e+256)
		tmp = Float64(sqrt(Float64(t_0 * Float64(-B_m))) / Float64(B_m * Float64(-B_m)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-101], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+256], N[(N[Sqrt[N[(t$95$0 * (-B$95$m)), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-101

   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 C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 21.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      2. lower-*.f64 20.5

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   10. Applied rewrites 20.5%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   if 2.0000000000000001e-101 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e256

   1. Initial program 43.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 10.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 10.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-1 \cdot {B}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(\color{blue}{B \cdot B}\right)} \]

      4. lower-*.f64 3.8

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\color{blue}{B \cdot B}} \]

   10. Applied rewrites 3.8%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-B \cdot B}} \]

   11. Taylor expanded in B around inf

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(B\right)\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       2. lower-neg.f64 14.5

          \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   13. Applied rewrites 14.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   if 2.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 3.2

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 3.2%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 3.9%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(-B\right)}}{B \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(-B\_m\right)}}{B\_m \cdot \left(-B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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) 2e-101)
     (/ (sqrt (* (* A -8.0) (* (* C F) (+ A A)))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+256)
       (/ (sqrt (* (* t_0 (* 2.0 F)) (- B_m))) (* B_m (- B_m)))
       (* 2.0 (* (sqrt (* A F)) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-101) {
		tmp = sqrt(((A * -8.0) * ((C * F) * (A + A)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+256) {
		tmp = sqrt(((t_0 * (2.0 * F)) * -B_m)) / (B_m * -B_m);
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-101)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(Float64(C * F) * Float64(A + A)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+256)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(-B_m))) / Float64(B_m * Float64(-B_m)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(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], 2e-101], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(C * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+256], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * (-B$95$m)), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-101

   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 C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 21.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right)} \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot F\right)} \cdot \left(A - -1 \cdot A\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A - \color{blue}{\left(\mathsf{neg}\left(A\right)\right)}\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      9. lower-neg.f64 18.0

         \[\leadsto \frac{\sqrt{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   10. Applied rewrites 18.0%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A - \left(-A\right)\right)\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 2.0000000000000001e-101 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e256

   1. Initial program 43.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 10.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 10.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-1 \cdot {B}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(\color{blue}{B \cdot B}\right)} \]

      4. lower-*.f64 3.8

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\color{blue}{B \cdot B}} \]

   10. Applied rewrites 3.8%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-B \cdot B}} \]

   11. Taylor expanded in B around inf

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(B\right)\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       2. lower-neg.f64 14.5

          \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   13. Applied rewrites 14.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   if 2.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 3.2

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 3.2%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 3.9%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 13.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(-B\right)}}{B \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 22.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(-B\_m\right)}}{B\_m \cdot \left(-B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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) 2e-101)
     (/ (sqrt (* -16.0 (* F (* C (* A A))))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+256)
       (/ (sqrt (* (* t_0 (* 2.0 F)) (- B_m))) (* B_m (- B_m)))
       (* 2.0 (* (sqrt (* A F)) (/ -1.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-101) {
		tmp = sqrt((-16.0 * (F * (C * (A * A))))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+256) {
		tmp = sqrt(((t_0 * (2.0 * F)) * -B_m)) / (B_m * -B_m);
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-101)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+256)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(-B_m))) / Float64(B_m * Float64(-B_m)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(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], 2e-101], N[(N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+256], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * (-B$95$m)), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-101

   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 C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 21.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      6. lower-*.f64 14.8

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   10. Applied rewrites 14.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 2.0000000000000001e-101 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e256

   1. Initial program 43.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 10.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 10.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-1 \cdot {B}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(\color{blue}{B \cdot B}\right)} \]

      4. lower-*.f64 3.8

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\color{blue}{B \cdot B}} \]

   10. Applied rewrites 3.8%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-B \cdot B}} \]

   11. Taylor expanded in B around inf

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(B\right)\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       2. lower-neg.f64 14.5

          \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   13. Applied rewrites 14.5%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-B\right)} \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-B \cdot B} \]

   if 2.0000000000000001e256 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 3.2

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 3.2%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 3.9%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(-B\right)}}{B \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 22.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{B\_m \cdot \left(-B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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) 2e-101)
   (/ (sqrt (* -16.0 (* F (* C (* A A))))) (- (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e+192)
     (/ (sqrt (* -2.0 (* F (* B_m (* B_m B_m))))) (* B_m (- B_m)))
     (* 2.0 (* (sqrt (* A F)) (/ -1.0 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 (pow(B_m, 2.0) <= 2e-101) {
		tmp = sqrt((-16.0 * (F * (C * (A * A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 2e+192) {
		tmp = sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / (B_m * -B_m);
	} else {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / 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 ((B_m ^ 2.0) <= 2e-101)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif ((B_m ^ 2.0) <= 2e+192)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / Float64(B_m * Float64(-B_m)));
	else
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-101], N[(N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+192], N[(N[Sqrt[N[(-2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-101

   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 C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 21.8

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      6. lower-*.f64 14.8

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   10. Applied rewrites 14.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 2.0000000000000001e-101 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000008e192

   1. Initial program 40.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 12.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 12.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 12.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-1 \cdot {B}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(\color{blue}{B \cdot B}\right)} \]

      4. lower-*.f64 4.0

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\color{blue}{B \cdot B}} \]

   10. Applied rewrites 4.0%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-B \cdot B}} \]

   11. Taylor expanded in B around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]

       3. cube-mult N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot \left(B \cdot B\right)\right)} \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{{B}^{2}}\right) \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot {B}^{2}\right)} \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       7. lower-*.f64 14.7

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{-B \cdot B} \]

   13. Applied rewrites 14.7%

       \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{-B \cdot B} \]

   if 2.00000000000000008e192 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 8.3%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 6.6

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 6.6%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 6.6%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 4.2%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot \left(-B\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 41.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \frac{1}{B\_m \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B\_m}, B\_m\right)\right)}\\ \end{array} \end{array} \]

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) 2e-76)
     (/ (sqrt (* (+ A A) (* t_0 (* 2.0 F)))) (- t_0))
     (*
      (* -2.0 (/ 1.0 (* B_m (sqrt 2.0))))
      (sqrt (* F (- A (fma 0.5 (/ (* A A) B_m) B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-76) {
		tmp = sqrt(((A + A) * (t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = (-2.0 * (1.0 / (B_m * sqrt(2.0)))) * sqrt((F * (A - fma(0.5, ((A * A) / B_m), B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-76)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-2.0 * Float64(1.0 / Float64(B_m * sqrt(2.0)))) * sqrt(Float64(F * Float64(A - fma(0.5, Float64(Float64(A * A) / B_m), B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(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], 2e-76], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(-2.0 * N[(1.0 / N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[(0.5 * N[(N[(A * A), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999985e-76

   1. Initial program 21.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 21.3

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 21.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 1.99999999999999985e-76 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 19.4%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 12.1

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 12.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 12.0%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around 0

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \left(B + \frac{1}{2} \cdot \frac{{A}^{2}}{B}\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 22.8%

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{fma}\left(0.5, \frac{A \cdot A}{B}, B\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 21.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ t_1 := \frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{t\_0}\\ \mathbf{if}\;A \leq -7.4 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \mathbf{elif}\;A \leq -2.35 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;A \leq 5.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 A (* C -4.0) (* B_m B_m))))
        (t_1 (/ (sqrt (* -16.0 (* F (* C (* A A))))) t_0)))
   (if (<= A -7.4e+153)
     (* 2.0 (* (sqrt (* A F)) (/ -1.0 B_m)))
     (if (<= A -2.35e-116)
       t_1
       (if (<= A 5.3e-124)
         (/ (sqrt (* -2.0 (* F (* B_m (* B_m B_m))))) t_0)
         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(A, (C * -4.0), (B_m * B_m));
	double t_1 = sqrt((-16.0 * (F * (C * (A * A))))) / t_0;
	double tmp;
	if (A <= -7.4e+153) {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	} else if (A <= -2.35e-116) {
		tmp = t_1;
	} else if (A <= 5.3e-124) {
		tmp = sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / t_0;
	} else {
		tmp = 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 = Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m)))
	t_1 = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))) / t_0)
	tmp = 0.0
	if (A <= -7.4e+153)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	elseif (A <= -2.35e-116)
		tmp = t_1;
	elseif (A <= 5.3e-124)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / t_0);
	else
		tmp = 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[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[A, -7.4e+153], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -2.35e-116], t$95$1, If[LessEqual[A, 5.3e-124], N[(N[Sqrt[N[(-2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -7.4000000000000005e153

   1. Initial program 1.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 0.0%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 1.7

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 1.7%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 1.7%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 7.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]

   if -7.4000000000000005e153 < A < -2.34999999999999997e-116 or 5.2999999999999997e-124 < A

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 13.6

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 13.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 13.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left({A}^{2} \cdot C\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      6. lower-*.f64 13.1

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   10. Applied rewrites 13.1%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if -2.34999999999999997e-116 < A < 5.2999999999999997e-124

   1. Initial program 36.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 8.5

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 8.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 8.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      3. cube-mult N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot \left(B \cdot B\right)\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{{B}^{2}}\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot {B}^{2}\right)} \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]

      7. lower-*.f64 9.8

         \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   10. Applied rewrites 9.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -7.4 \cdot 10^{+153}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \mathbf{elif}\;A \leq -2.35 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 5.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 12.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B\_m \cdot \left(B\_m \cdot B\_m\right)\right)\right)}}{B\_m \cdot \left(-B\_m\right)}\\ \end{array} \end{array} \]

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 (<= A -3.2e-115)
   (* 2.0 (* (sqrt (* A F)) (/ -1.0 B_m)))
   (/ (sqrt (* -2.0 (* F (* B_m (* B_m B_m))))) (* B_m (- 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 (A <= -3.2e-115) {
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	} else {
		tmp = sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / (B_m * -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 (a <= (-3.2d-115)) then
        tmp = 2.0d0 * (sqrt((a * f)) * ((-1.0d0) / b_m))
    else
        tmp = sqrt(((-2.0d0) * (f * (b_m * (b_m * b_m))))) / (b_m * -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 (A <= -3.2e-115) {
		tmp = 2.0 * (Math.sqrt((A * F)) * (-1.0 / B_m));
	} else {
		tmp = Math.sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / (B_m * -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 A <= -3.2e-115:
		tmp = 2.0 * (math.sqrt((A * F)) * (-1.0 / B_m))
	else:
		tmp = math.sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / (B_m * -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 (A <= -3.2e-115)
		tmp = Float64(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(F * Float64(B_m * Float64(B_m * B_m))))) / Float64(B_m * 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 (A <= -3.2e-115)
		tmp = 2.0 * (sqrt((A * F)) * (-1.0 / B_m));
	else
		tmp = sqrt((-2.0 * (F * (B_m * (B_m * B_m))))) / (B_m * -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[A, -3.2e-115], N[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * N[(B$95$m * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.2e-115

   1. Initial program 19.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. flip-+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. neg-sub0 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 18.1%

      \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      9. lower--.f64 N/A

         \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

      11. +-commutative N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

      12. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

      14. unpow2 N/A

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

      15. lower-*.f64 6.1

          \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Applied rewrites 6.1%

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 6.1%

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

   10. Taylor expanded in A around -inf

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 5.6%

       \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]

   if -3.2e-115 < A

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{1} \cdot A\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{A}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-+.f64 6.2

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 6.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 6.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-1 \cdot {B}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{neg}\left({B}^{2}\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{neg}\left(\color{blue}{B \cdot B}\right)} \]

      4. lower-*.f64 1.0

         \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{-\color{blue}{B \cdot B}} \]

   10. Applied rewrites 1.0%

       \[\leadsto \frac{\sqrt{\left(A + A\right) \cdot \left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{-B \cdot B}} \]

   11. Taylor expanded in B around inf

       \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{neg}\left(B \cdot B\right)} \]

       3. cube-mult N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot \left(B \cdot B\right)\right)} \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       4. unpow2 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{{B}^{2}}\right) \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\color{blue}{\left(B \cdot {B}^{2}\right)} \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       6. unpow2 N/A

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{\mathsf{neg}\left(B \cdot B\right)} \]

       7. lower-*.f64 5.8

          \[\leadsto \frac{\sqrt{-2 \cdot \left(\left(B \cdot \color{blue}{\left(B \cdot B\right)}\right) \cdot F\right)}}{-B \cdot B} \]

   13. Applied rewrites 5.8%

       \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left(\left(B \cdot \left(B \cdot B\right)\right) \cdot F\right)}}}{-B \cdot B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{B \cdot \left(-B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 9.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ 2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{-1}{B\_m}\right) \end{array} \]

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
 (* 2.0 (* (sqrt (* A F)) (/ -1.0 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 2.0 * (sqrt((A * F)) * (-1.0 / 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 = 2.0d0 * (sqrt((a * f)) * ((-1.0d0) / 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 2.0 * (Math.sqrt((A * F)) * (-1.0 / 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 2.0 * (math.sqrt((A * F)) * (-1.0 / 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(2.0 * Float64(sqrt(Float64(A * F)) * Float64(-1.0 / 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 = 2.0 * (sqrt((A * F)) * (-1.0 / 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[(2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-lft-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(0 + \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. flip-+ N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)} \cdot \sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

4. Applied rewrites 19.3%

   \[\leadsto \frac{-\color{blue}{\frac{-\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-2 \cdot \left(\frac{1}{B \cdot \sqrt{2}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(-2 \cdot \color{blue}{\frac{1}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \frac{1}{\color{blue}{B \cdot \sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \color{blue}{\sqrt{2}}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)} \]

   11. +-commutative N/A

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{{B}^{2} + {A}^{2}}}\right)} \]

   12. unpow2 N/A

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(B, B, {A}^{2}\right)}}\right)} \]

   14. unpow2 N/A

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   15. lower-*.f64 8.9

       \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

7. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 8.7%

   \[\leadsto \left(-2 \cdot \frac{1}{B \cdot \sqrt{2}}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}}\right)} \]

10. Taylor expanded in A around -inf

    \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{B}\right)} \]
11. Step-by-step derivation

12. Applied rewrites 2.7%

    \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{-1}{B}\right)} \]
13. Add Preprocessing

Alternative 22: 1.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{2}{\frac{B\_m}{F}}} \end{array} \]

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 (/ B_m 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((2.0 / (B_m / 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((2.0d0 / (b_m / 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((2.0 / (B_m / 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((2.0 / (B_m / 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 sqrt(Float64(2.0 / Float64(B_m / 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((2.0 / (B_m / 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[Sqrt[N[(2.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{2}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \sqrt{2}}\right)\right) \]

   10. lower-sqrt.f64 1.8

       \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \color{blue}{\sqrt{2}}\right) \]

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 1.8%

   \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]
8. Step-by-step derivation

9. Applied rewrites 1.8%

   \[\leadsto \sqrt{\frac{2}{\frac{B}{F}}} \]
10. Add Preprocessing

Alternative 23: 1.6% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* 2.0 (/ F B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 * (F / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 * Float64(F / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.9%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{2}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \sqrt{2}}\right)\right) \]

   10. lower-sqrt.f64 1.8

       \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \color{blue}{\sqrt{2}}\right) \]

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 1.8%

   \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]

8. Final simplification 1.8%

   \[\leadsto \sqrt{2 \cdot \frac{F}{B}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))