ABCF->ab-angle b

Percentage Accurate: 19.0% → 40.8%

Time: 20.5s

Alternatives: 16

Speedup: 10.2×

• could not determine a ground truth

  1. A = 4.213318523369787e+261
  2. B = -285858120447955070.0
  3. C = 2.1972667141680205e-225
  4. F = -5.608147516946704e+33

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 40.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B}^{2}\\ t_2 := {B}^{2} - t\_0\\ t_3 := 2 \cdot \left(t\_2 \cdot F\right)\\ t_4 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{1}{A - C}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(-8 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B 2.0)))
        (t_2 (- (pow B 2.0) t_0))
        (t_3 (* 2.0 (* t_2 F)))
        (t_4
         (/
          (sqrt (* t_3 (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_1)))
   (if (<= t_4 (- INFINITY))
     (/ (* (* -2.0 (sqrt (* A F))) (sqrt (fma C (* A -4.0) (* B B)))) t_2)
     (if (<= t_4 -1e-217)
       (/
        (sqrt
         (*
          t_3
          (fma
           (* (+ A C) (- A C))
           (/ 1.0 (- A C))
           (- (sqrt (fma (- A C) (- A C) (* B B)))))))
        t_1)
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_3 (+ A (fma (/ (* B B) C) -0.5 A)))) t_1)
         (/ (* (sqrt (* 2.0 (* F (* -8.0 (* A C))))) (sqrt A)) t_1))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B, 2.0);
	double t_2 = pow(B, 2.0) - t_0;
	double t_3 = 2.0 * (t_2 * F);
	double t_4 = sqrt((t_3 * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((-2.0 * sqrt((A * F))) * sqrt(fma(C, (A * -4.0), (B * B)))) / t_2;
	} else if (t_4 <= -1e-217) {
		tmp = sqrt((t_3 * fma(((A + C) * (A - C)), (1.0 / (A - C)), -sqrt(fma((A - C), (A - C), (B * B)))))) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (A + fma(((B * B) / C), -0.5, A)))) / t_1;
	} else {
		tmp = (sqrt((2.0 * (F * (-8.0 * (A * C))))) * sqrt(A)) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B ^ 2.0))
	t_2 = Float64((B ^ 2.0) - t_0)
	t_3 = Float64(2.0 * Float64(t_2 * F))
	t_4 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_1)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0 * sqrt(Float64(A * F))) * sqrt(fma(C, Float64(A * -4.0), Float64(B * B)))) / t_2);
	elseif (t_4 <= -1e-217)
		tmp = Float64(sqrt(Float64(t_3 * fma(Float64(Float64(A + C) * Float64(A - C)), Float64(1.0 / Float64(A - C)), Float64(-sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_3 * Float64(A + fma(Float64(Float64(B * B) / C), -0.5, A)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(-8.0 * Float64(A * C))))) * sqrt(A)) / t_1);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$3 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(-2.0 * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-217], N[(N[Sqrt[N[(t$95$3 * N[(N[(N[(A + C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(A - C), $MachinePrecision]), $MachinePrecision] + (-N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$3 * N[(A + N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / t$95$1), $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 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.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. Simplified 20.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.5

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

   8. Simplified 20.5%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

   10. Applied egg-rr 24.8%

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

   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))) < -1.00000000000000008e-217

   1. Initial program 99.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. sqr-neg N/A

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

      6. difference-of-squares N/A

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

      7. sub-neg N/A

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

      8. flip-- N/A

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

      9. sqr-neg N/A

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

      10. sub-neg N/A

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

      11. flip-+ N/A

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

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

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

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

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

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

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

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

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

      16. --lowering--.f64 N/A

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

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

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

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

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

   4. Applied egg-rr 99.0%

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

   if -1.00000000000000008e-217 < (/.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 19.6%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 N/A

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

      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. accelerator-lowering-fma.f64 N/A

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

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

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

      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. *-lowering-*.f64 29.6

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

   5. Simplified 29.6%

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

   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. 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. +-lowering-+.f64 5.6

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

   5. Simplified 5.6%

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

   6. Taylor expanded in B around 0

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

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

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

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

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

      3. *-lowering-*.f64 5.3

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

   8. Simplified 5.3%

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

      1. count-2 N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

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

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

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

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

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

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

      13. pow1/2 N/A

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

      14. sqrt-lowering-sqrt.f64 7.9

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

   10. Applied egg-rr 7.9%

       \[\leadsto \frac{-\color{blue}{\sqrt{\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot 2} \cdot \sqrt{A}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{1}{A - C}, -\sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(-8 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 40.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B}^{2}\\ t_2 := {B}^{2} - t\_0\\ t_3 := 2 \cdot \left(t\_2 \cdot F\right)\\ t_4 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1}\\ t_5 := \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{t\_5}{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_5\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(-8 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B 2.0)))
        (t_2 (- (pow B 2.0) t_0))
        (t_3 (* 2.0 (* t_2 F)))
        (t_4
         (/
          (sqrt (* t_3 (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          t_1))
        (t_5 (fma B B (* -4.0 (* A C)))))
   (if (<= t_4 (- INFINITY))
     (/ (* (* -2.0 (sqrt (* A F))) (sqrt (fma C (* A -4.0) (* B B)))) t_2)
     (if (<= t_4 -1e-217)
       (/
        -1.0
        (/
         t_5
         (sqrt
          (*
           (* (* 2.0 F) t_5)
           (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_3 (+ A (fma (/ (* B B) C) -0.5 A)))) t_1)
         (/ (* (sqrt (* 2.0 (* F (* -8.0 (* A C))))) (sqrt A)) t_1))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B, 2.0);
	double t_2 = pow(B, 2.0) - t_0;
	double t_3 = 2.0 * (t_2 * F);
	double t_4 = sqrt((t_3 * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double t_5 = fma(B, B, (-4.0 * (A * C)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((-2.0 * sqrt((A * F))) * sqrt(fma(C, (A * -4.0), (B * B)))) / t_2;
	} else if (t_4 <= -1e-217) {
		tmp = -1.0 / (t_5 / sqrt((((2.0 * F) * t_5) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (A + fma(((B * B) / C), -0.5, A)))) / t_1;
	} else {
		tmp = (sqrt((2.0 * (F * (-8.0 * (A * C))))) * sqrt(A)) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B ^ 2.0))
	t_2 = Float64((B ^ 2.0) - t_0)
	t_3 = Float64(2.0 * Float64(t_2 * F))
	t_4 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_1)
	t_5 = fma(B, B, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0 * sqrt(Float64(A * F))) * sqrt(fma(C, Float64(A * -4.0), Float64(B * B)))) / t_2);
	elseif (t_4 <= -1e-217)
		tmp = Float64(-1.0 / Float64(t_5 / sqrt(Float64(Float64(Float64(2.0 * F) * t_5) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_3 * Float64(A + fma(Float64(Float64(B * B) / C), -0.5, A)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(-8.0 * Float64(A * C))))) * sqrt(A)) / t_1);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$3 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(-2.0 * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -1e-217], N[(-1.0 / N[(t$95$5 / N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$3 * N[(A + N[(N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(-8.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] / t$95$1), $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 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.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. Simplified 20.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.5

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

   8. Simplified 20.5%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

   10. Applied egg-rr 24.8%

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

   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))) < -1.00000000000000008e-217

   1. Initial program 99.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 99.0%

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

   if -1.00000000000000008e-217 < (/.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 19.6%

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

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 N/A

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

      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. accelerator-lowering-fma.f64 N/A

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

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

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

      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. *-lowering-*.f64 29.6

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

   5. Simplified 29.6%

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

   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. 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. +-lowering-+.f64 5.6

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

   5. Simplified 5.6%

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

   6. Taylor expanded in B around 0

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

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

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

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

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

      3. *-lowering-*.f64 5.3

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

   8. Simplified 5.3%

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

      1. count-2 N/A

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

      2. associate-*r* N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

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

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

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

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

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

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

      13. pow1/2 N/A

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

      14. sqrt-lowering-sqrt.f64 7.9

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

   10. Applied egg-rr 7.9%

       \[\leadsto \frac{-\color{blue}{\sqrt{\left(\left(-8 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot 2} \cdot \sqrt{A}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(-8 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{A}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := {B}^{2} - t\_2\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ t_5 := \mathsf{fma}\left(B, B, t\_0\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{t\_1}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{t\_5}{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_5\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{-2 \cdot \sqrt{A \cdot \left(F \cdot t\_0\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C)))
        (t_1 (fma C (* A -4.0) (* B B)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- (pow B 2.0) t_2))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B 2.0))))
        (t_5 (fma B B t_0)))
   (if (<= t_4 (- INFINITY))
     (/ (* (* -2.0 (sqrt (* A F))) (sqrt t_1)) t_3)
     (if (<= t_4 -1e-217)
       (/
        -1.0
        (/
         t_5
         (sqrt
          (*
           (* (* 2.0 F) t_5)
           (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))))
       (/ 1.0 (/ t_1 (* -2.0 (sqrt (* A (* F t_0))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double t_1 = fma(C, (A * -4.0), (B * B));
	double t_2 = (4.0 * A) * C;
	double t_3 = pow(B, 2.0) - t_2;
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B, 2.0));
	double t_5 = fma(B, B, t_0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((-2.0 * sqrt((A * F))) * sqrt(t_1)) / t_3;
	} else if (t_4 <= -1e-217) {
		tmp = -1.0 / (t_5 / sqrt((((2.0 * F) * t_5) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))));
	} else {
		tmp = 1.0 / (t_1 / (-2.0 * sqrt((A * (F * t_0)))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	t_1 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64((B ^ 2.0) - t_2)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B ^ 2.0)))
	t_5 = fma(B, B, t_0)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-2.0 * sqrt(Float64(A * F))) * sqrt(t_1)) / t_3);
	elseif (t_4 <= -1e-217)
		tmp = Float64(-1.0 / Float64(t_5 / sqrt(Float64(Float64(Float64(2.0 * F) * t_5) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))));
	else
		tmp = Float64(1.0 / Float64(t_1 / Float64(-2.0 * sqrt(Float64(A * Float64(F * t_0))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B * B + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(-2.0 * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -1e-217], N[(-1.0 / N[(t$95$5 / N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 / N[(-2.0 * N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.4%

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

   3. Taylor expanded in 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. +-lowering-+.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. Simplified 20.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.5

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

   8. Simplified 20.5%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. pow2 N/A

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

      14. associate-*l* N/A

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

   10. Applied egg-rr 24.8%

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

   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))) < -1.00000000000000008e-217

   1. Initial program 99.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 99.0%

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

   if -1.00000000000000008e-217 < (/.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 7.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.4

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

   5. Simplified 11.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 11.3%

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

   11. Taylor expanded in C around inf

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

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

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

       2. *-lowering-*.f64 12.4

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

   13. Simplified 12.4%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_4 := \mathsf{fma}\left(B, B, t\_0\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{t\_3}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{t\_3}\right)}}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{t\_4}{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_4\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_3}{-2 \cdot \sqrt{A \cdot \left(F \cdot t\_0\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0))))
        (t_3 (fma C (* A -4.0) (* B B)))
        (t_4 (fma B B t_0)))
   (if (<= t_2 (- INFINITY))
     (/ 1.0 (/ t_3 (* -2.0 (* (sqrt (* A F)) (sqrt t_3)))))
     (if (<= t_2 -1e-217)
       (/
        -1.0
        (/
         t_4
         (sqrt
          (*
           (* (* 2.0 F) t_4)
           (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))))
       (/ 1.0 (/ t_3 (* -2.0 (sqrt (* A (* F t_0))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double t_3 = fma(C, (A * -4.0), (B * B));
	double t_4 = fma(B, B, t_0);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 1.0 / (t_3 / (-2.0 * (sqrt((A * F)) * sqrt(t_3))));
	} else if (t_2 <= -1e-217) {
		tmp = -1.0 / (t_4 / sqrt((((2.0 * F) * t_4) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))));
	} else {
		tmp = 1.0 / (t_3 / (-2.0 * sqrt((A * (F * t_0)))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_4 = fma(B, B, t_0)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(t_3 / Float64(-2.0 * Float64(sqrt(Float64(A * F)) * sqrt(t_3)))));
	elseif (t_2 <= -1e-217)
		tmp = Float64(-1.0 / Float64(t_4 / sqrt(Float64(Float64(Float64(2.0 * F) * t_4) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))));
	else
		tmp = Float64(1.0 / Float64(t_3 / Float64(-2.0 * sqrt(Float64(A * Float64(F * t_0))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $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, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(B * B + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(1.0 / N[(t$95$3 / N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-217], N[(-1.0 / N[(t$95$4 / N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$3 / N[(-2.0 * N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.4%

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

   3. Taylor expanded in 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. +-lowering-+.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. Simplified 20.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.5

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

   8. Simplified 20.5%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 20.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

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

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

       6. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 24.8

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

   12. Applied egg-rr 24.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000008e-217

   1. Initial program 99.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 99.0%

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

   if -1.00000000000000008e-217 < (/.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 7.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.4

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

   5. Simplified 11.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 11.3%

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

   11. Taylor expanded in C around inf

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

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

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

       2. *-lowering-*.f64 12.4

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

   13. Simplified 12.4%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{t\_0}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{t\_0}\right)}}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, t\_1\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{-2 \cdot \sqrt{A \cdot \left(F \cdot t\_1\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B)))
        (t_1 (* -4.0 (* A C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B 2.0)))))
   (if (<= t_3 (- INFINITY))
     (/ 1.0 (/ t_0 (* -2.0 (* (sqrt (* A F)) (sqrt t_0)))))
     (if (<= t_3 -1e-217)
       (/
        (sqrt
         (*
          (* (* 2.0 F) (fma B B t_1))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B))))))
        (fma B (- B) t_2))
       (/ 1.0 (/ t_0 (* -2.0 (sqrt (* A (* F t_1))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double t_1 = -4.0 * (A * C);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 1.0 / (t_0 / (-2.0 * (sqrt((A * F)) * sqrt(t_0))));
	} else if (t_3 <= -1e-217) {
		tmp = sqrt((((2.0 * F) * fma(B, B, t_1)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))) / fma(B, -B, t_2);
	} else {
		tmp = 1.0 / (t_0 / (-2.0 * sqrt((A * (F * t_1)))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_1 = Float64(-4.0 * Float64(A * C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(t_0 / Float64(-2.0 * Float64(sqrt(Float64(A * F)) * sqrt(t_0)))));
	elseif (t_3 <= -1e-217)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * fma(B, B, t_1)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B)))))) / fma(B, Float64(-B), t_2));
	else
		tmp = Float64(1.0 / Float64(t_0 / Float64(-2.0 * sqrt(Float64(A * Float64(F * t_1))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(1.0 / N[(t$95$0 / N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-217], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(B * B + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + t$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 / N[(-2.0 * N[Sqrt[N[(A * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 3.4%

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

   3. Taylor expanded in 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. +-lowering-+.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. Simplified 20.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.5

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

   8. Simplified 20.5%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 20.5%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

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

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

       6. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 24.8

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

   12. Applied egg-rr 24.8%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.00000000000000008e-217

   1. Initial program 99.0%

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

      1. remove-double-neg N/A

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

      2. frac-2neg N/A

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

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

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

   4. Applied egg-rr 99.0%

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

   if -1.00000000000000008e-217 < (/.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 7.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.4

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

   5. Simplified 11.4%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 11.3

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

   8. Simplified 11.3%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 11.3%

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

   11. Taylor expanded in C around inf

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

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

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

       2. *-lowering-*.f64 12.4

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

   13. Simplified 12.4%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{t\_0}\right)}}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -4e+200)
     (/ 1.0 (/ t_0 (* -2.0 (* (sqrt (* A F)) (sqrt t_0)))))
     (if (<= t_2 -2e-162)
       (*
        (sqrt
         (/
          (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
          (fma (* A C) -4.0 (* B B))))
        (- (sqrt 2.0)))
       (/ 1.0 (/ t_0 (* -2.0 (sqrt (* A (* F (* -4.0 (* A C))))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -4e+200) {
		tmp = 1.0 / (t_0 / (-2.0 * (sqrt((A * F)) * sqrt(t_0))));
	} else if (t_2 <= -2e-162) {
		tmp = sqrt(((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma((A * C), -4.0, (B * B)))) * -sqrt(2.0);
	} else {
		tmp = 1.0 / (t_0 / (-2.0 * sqrt((A * (F * (-4.0 * (A * C)))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -4e+200)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-2.0 * Float64(sqrt(Float64(A * F)) * sqrt(t_0)))));
	elseif (t_2 <= -2e-162)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(Float64(A * C), -4.0, Float64(B * B)))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(1.0 / Float64(t_0 / Float64(-2.0 * sqrt(Float64(A * Float64(F * Float64(-4.0 * Float64(A * C))))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $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, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+200], N[(1.0 / N[(t$95$0 / N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-162], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[(t$95$0 / N[(-2.0 * N[Sqrt[N[(A * N[(F * N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.9999999999999999e200

   1. Initial program 5.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 20.2

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

   5. Simplified 20.2%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.3

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

   8. Simplified 20.3%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 20.2%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

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

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

       6. pow1/2 N/A

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

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

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

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

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

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

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

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

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

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

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

       12. *-lowering-*.f64 24.5

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

   12. Applied egg-rr 24.5%

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

   if -3.9999999999999999e200 < (/.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))) < -1.99999999999999991e-162

   1. Initial program 98.9%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 95.1%

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

   if -1.99999999999999991e-162 < (/.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 8.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.3

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

   5. Simplified 11.3%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 11.2

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

   8. Simplified 11.2%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 11.2%

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

   11. Taylor expanded in C around inf

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

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

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

       2. *-lowering-*.f64 12.3

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

   13. Simplified 12.3%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{+200}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, t\_0\right)}}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + A\right)}}{-t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B 2.0)))))
   (if (<= t_2 -5e-88)
     (* -2.0 (sqrt (/ (* A F) (fma B B t_0))))
     (if (<= t_2 -2e-162)
       (/ (sqrt (* -2.0 (* F (* B (* B B))))) (fma B (- B) t_1))
       (/ (sqrt (* (* -8.0 (* A (* C F))) (+ A A))) (- t_0))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0));
	double tmp;
	if (t_2 <= -5e-88) {
		tmp = -2.0 * sqrt(((A * F) / fma(B, B, t_0)));
	} else if (t_2 <= -2e-162) {
		tmp = sqrt((-2.0 * (F * (B * (B * B))))) / fma(B, -B, t_1);
	} else {
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + A))) / -t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -5e-88)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(B, B, t_0))));
	elseif (t_2 <= -2e-162)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(F * Float64(B * Float64(B * B))))) / fma(B, Float64(-B), t_1));
	else
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(A * Float64(C * F))) * Float64(A + A))) / Float64(-t_0));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $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, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-88], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-162], N[(N[Sqrt[N[(-2.0 * N[(F * N[(B * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-88

   1. Initial program 36.8%

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

   3. Taylor expanded in C around 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. +-lowering-+.f64 19.5

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

   5. Simplified 19.5%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 19.5

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

   8. Simplified 19.5%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 19.5%

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

   11. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-lowering-*.f64 19.8

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

   13. Simplified 19.8%

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

   if -5.00000000000000009e-88 < (/.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))) < -1.99999999999999991e-162

   1. Initial program 99.7%

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

      1. remove-double-neg N/A

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

      2. frac-2neg N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]

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

         \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]

      3. cube-mult N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 41.4

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

   7. Simplified 41.4%

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

   if -1.99999999999999991e-162 < (/.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 8.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.3

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

   5. Simplified 11.3%

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

   6. Taylor expanded in B around 0

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

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

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

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

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

      3. *-lowering-*.f64 9.6

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

   8. Simplified 9.6%

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

   9. Taylor expanded in B around 0

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

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

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

       2. *-lowering-*.f64 10.7

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

   11. Simplified 10.7%

       \[\leadsto \frac{-\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + A\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-88}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + A\right)}}{--4 \cdot \left(A \cdot C\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B}^{2}} \leq -2 \cdot 10^{-162}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, t\_0\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + A\right)}}{-t\_0}\\ \end{array} \end{array} \]

FPCore

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

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double t_1 = (4.0 * A) * C;
	double tmp;
	if ((sqrt(((2.0 * ((pow(B, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B, 2.0))) <= -2e-162) {
		tmp = -2.0 * sqrt(((A * F) / fma(B, B, t_0)));
	} else {
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + A))) / -t_0;
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-162], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.99999999999999991e-162

   1. Initial program 44.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 19.9

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

   5. Simplified 19.9%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 20.0

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

   8. Simplified 20.0%

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

      1. clear-num N/A

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

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

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

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

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

   10. Applied egg-rr 20.0%

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

   11. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-lowering-*.f64 20.3

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

   13. Simplified 20.3%

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

   if -1.99999999999999991e-162 < (/.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 8.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 11.3

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

   5. Simplified 11.3%

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

   6. Taylor expanded in B around 0

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

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

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

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

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

      3. *-lowering-*.f64 9.6

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

   8. Simplified 9.6%

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

   9. Taylor expanded in B around 0

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

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

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

       2. *-lowering-*.f64 10.7

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

   11. Simplified 10.7%

       \[\leadsto \frac{-\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + A\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9%

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

Alternative 9: 24.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B}^{2} \leq 2.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C))))
   (if (<= (pow B 2.0) 5e-64)
     (/ (* -2.0 (sqrt (* (* A F) (fma B B t_0)))) t_0)
     (if (<= (pow B 2.0) 2.3e+114)
       (/ (sqrt (* -2.0 (* F (* B (* B B))))) (fma B (- B) (* (* 4.0 A) C)))
       (* -2.0 (* (sqrt (* A F)) (/ 1.0 B)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double tmp;
	if (pow(B, 2.0) <= 5e-64) {
		tmp = (-2.0 * sqrt(((A * F) * fma(B, B, t_0)))) / t_0;
	} else if (pow(B, 2.0) <= 2.3e+114) {
		tmp = sqrt((-2.0 * (F * (B * (B * B))))) / fma(B, -B, ((4.0 * A) * C));
	} else {
		tmp = -2.0 * (sqrt((A * F)) * (1.0 / B));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-64)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(A * F) * fma(B, B, t_0)))) / t_0);
	elseif ((B ^ 2.0) <= 2.3e+114)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(F * Float64(B * Float64(B * B))))) / fma(B, Float64(-B), Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-64], N[(N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2.3e+114], N[(N[Sqrt[N[(-2.0 * N[(F * N[(B * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B * (-B) + N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000033e-64

   1. Initial program 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 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. +-lowering-+.f64 26.6

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

   5. Simplified 26.6%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 25.7

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

   8. Simplified 25.7%

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

   9. Taylor expanded in B around 0

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

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

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

       2. *-lowering-*.f64 25.5

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

   11. Simplified 25.5%

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

   if 5.00000000000000033e-64 < (pow.f64 B #s(literal 2 binary64)) < 2.3e114

   1. Initial program 45.7%

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

      1. remove-double-neg N/A

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

      2. frac-2neg N/A

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

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

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

   4. Applied egg-rr 45.7%

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

   5. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \frac{\sqrt{\color{blue}{-2 \cdot \left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]

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

         \[\leadsto \frac{\sqrt{-2 \cdot \color{blue}{\left({B}^{3} \cdot F\right)}}}{\mathsf{fma}\left(B, \mathsf{neg}\left(B\right), C \cdot \left(A \cdot 4\right)\right)} \]

      3. cube-mult N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 18.7

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

   7. Simplified 18.7%

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

   if 2.3e114 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 2.8

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

   5. Simplified 2.8%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 2.8

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

   8. Simplified 2.8%

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

   9. Taylor expanded in A around 0

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

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

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

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

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

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

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

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

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

       5. /-lowering-/.f64 6.0

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

   11. Simplified 6.0%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-64}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(F \cdot \left(B \cdot \left(B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, -B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq 4.4 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{A \cdot \left(F \cdot t\_0\right)} \cdot \frac{-2}{t\_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B))))
   (if (<= B 4.4e-32)
     (* (sqrt (* A (* F t_0))) (/ -2.0 t_0))
     (if (<= B 9e+156)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double tmp;
	if (B <= 4.4e-32) {
		tmp = sqrt((A * (F * t_0))) * (-2.0 / t_0);
	} else if (B <= 9e+156) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= 4.4e-32)
		tmp = Float64(sqrt(Float64(A * Float64(F * t_0))) * Float64(-2.0 / t_0));
	elseif (B <= 9e+156)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4.4e-32], N[(N[Sqrt[N[(A * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 9e+156], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.4e-32

   1. Initial program 19.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. +-lowering-+.f64 16.9

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

   5. Simplified 16.9%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 16.3

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

   8. Simplified 16.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   10. Applied egg-rr 16.8%

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

   if 4.4e-32 < B < 9.00000000000000061e156

   1. Initial program 36.2%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

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

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

   5. Simplified 34.7%

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

   if 9.00000000000000061e156 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. +-lowering-+.f64 0.0

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

   5. Simplified 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 0.0

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 0.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right)}}}} \]

   11. Taylor expanded in C around 0

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       6. *-lowering-*.f64 6.6

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   13. Simplified 6.6%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{A \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)} \cdot \frac{-2}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 2.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, t\_0\right)}}{t\_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+156}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C))))
   (if (<= B 2.9e-43)
     (/ (* -2.0 (sqrt (* (* A F) (fma B B t_0)))) t_0)
     (if (<= B 9e+156)
       (/ (* (sqrt 2.0) (sqrt (* F (- A (sqrt (fma B B (* A A))))))) (- B))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double tmp;
	if (B <= 2.9e-43) {
		tmp = (-2.0 * sqrt(((A * F) * fma(B, B, t_0)))) / t_0;
	} else if (B <= 9e+156) {
		tmp = (sqrt(2.0) * sqrt((F * (A - sqrt(fma(B, B, (A * A))))))) / -B;
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if (B <= 2.9e-43)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(A * F) * fma(B, B, t_0)))) / t_0);
	elseif (B <= 9e+156)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A))))))) / Float64(-B));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 2.9e-43], N[(N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 9e+156], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.9000000000000001e-43

   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. 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. +-lowering-+.f64 16.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 16.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 16.4

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 16.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Taylor expanded in B around 0

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

       2. *-lowering-*.f64 16.3

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   11. Simplified 16.3%

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   if 2.9000000000000001e-43 < B < 9.00000000000000061e156

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 34.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}}{-B}} \]

   if 9.00000000000000061e156 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-lowering-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 0.0

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 0.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right)}}}} \]

   11. Taylor expanded in C around 0

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       6. *-lowering-*.f64 6.6

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   13. Simplified 6.6%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, t\_0\right)}}{t\_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* -4.0 (* A C))))
   (if (<= B 1.8e-42)
     (/ (* -2.0 (sqrt (* (* A F) (fma B B t_0)))) t_0)
     (if (<= B 9e+156)
       (* (sqrt (* F (- A (sqrt (fma B B (* A A)))))) (- (/ (sqrt 2.0) B)))
       (* -2.0 (/ (sqrt (* A F)) B))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = -4.0 * (A * C);
	double tmp;
	if (B <= 1.8e-42) {
		tmp = (-2.0 * sqrt(((A * F) * fma(B, B, t_0)))) / t_0;
	} else if (B <= 9e+156) {
		tmp = sqrt((F * (A - sqrt(fma(B, B, (A * A)))))) * -(sqrt(2.0) / B);
	} else {
		tmp = -2.0 * (sqrt((A * F)) / B);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if (B <= 1.8e-42)
		tmp = Float64(Float64(-2.0 * sqrt(Float64(Float64(A * F) * fma(B, B, t_0)))) / t_0);
	elseif (B <= 9e+156)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / B)));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.8e-42], N[(N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] * N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 9e+156], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.8000000000000001e-42

   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. 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. +-lowering-+.f64 16.9

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 16.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 16.4

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 16.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Taylor expanded in B around 0

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

       2. *-lowering-*.f64 16.3

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \color{blue}{\left(A \cdot C\right)}} \]

   11. Simplified 16.3%

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

   if 1.8000000000000001e-42 < B < 9.00000000000000061e156

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   4. Applied egg-rr 35.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, C \cdot \left(A \cdot 4\right)\right)}} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      3. *-lowering-*.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. /-lowering-/.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. sqrt-lowering-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. sqrt-lowering-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. *-lowering-*.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. --lowering--.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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 34.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, \color{blue}{A \cdot A}\right)}\right)} \]

   7. Simplified 34.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)}} \]

   if 9.00000000000000061e156 < B

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-lowering-+.f64 0.0

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 0.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 0.0

          \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 0.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   10. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right)}}}} \]

   11. Taylor expanded in C around 0

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

       2. *-rgt-identity N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

       5. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

       6. *-lowering-*.f64 6.6

          \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

   13. Simplified 6.6%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 18.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 19.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (* -2.0 (sqrt (/ (* A F) (fma B B (* -4.0 (* A C)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -2.0 * sqrt(((A * F) / fma(B, B, (-4.0 * (A * C)))));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(B, B, Float64(-4.0 * Float64(A * C))))))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 14.1

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Simplified 14.1%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. associate-*r* N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. cancel-sign-sub-inv N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. metadata-eval N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   11. *-lowering-*.f64 14.1

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

8. Simplified 14.1%

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

10. Applied egg-rr 14.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right)}}}} \]

11. Taylor expanded in F around 0

    \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{A \cdot F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{A \cdot F}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{A \cdot F}}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \]

    5. +-commutative N/A

       \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{{B}^{2} + -4 \cdot \left(A \cdot C\right)}}} \]

    6. unpow2 N/A

       \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{B \cdot B} + -4 \cdot \left(A \cdot C\right)}} \]

    7. accelerator-lowering-fma.f64 N/A

       \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

    8. *-lowering-*.f64 N/A

       \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}} \]

    9. *-lowering-*.f64 9.5

       \[\leadsto -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}} \]

13. Simplified 9.5%

    \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]
14. Add Preprocessing

Alternative 14: 5.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right) \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (* (sqrt (* A F)) (/ 1.0 B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) * (1.0 / B));
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (sqrt((a * f)) * (1.0d0 / b))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((A * F)) * (1.0 / B));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) * (1.0 / B))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(A * F)) * Float64(1.0 / B)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) * (1.0 / B));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 14.1

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Simplified 14.1%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. associate-*r* N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. cancel-sign-sub-inv N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. metadata-eval N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   11. *-lowering-*.f64 14.1

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

8. Simplified 14.1%

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

9. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto -2 \cdot \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]

    3. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto -2 \cdot \left(\color{blue}{\sqrt{A \cdot F}} \cdot \frac{1}{B}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto -2 \cdot \left(\sqrt{\color{blue}{A \cdot F}} \cdot \frac{1}{B}\right) \]

    5. /-lowering-/.f64 3.9

       \[\leadsto -2 \cdot \left(\sqrt{A \cdot F} \cdot \color{blue}{\frac{1}{B}}\right) \]

11. Simplified 3.9%

    \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
12. Add Preprocessing

Alternative 15: 5.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ -2 \cdot \frac{\sqrt{A \cdot F}}{B} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* -2.0 (/ (sqrt (* A F)) B)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -2.0 * (sqrt((A * F)) / B);
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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
    code = (-2.0d0) * (sqrt((a * f)) / b)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return -2.0 * (Math.sqrt((A * F)) / B);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return -2.0 * (math.sqrt((A * F)) / B)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = -2.0 * (sqrt((A * F)) / B);
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-lowering-+.f64 14.1

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

5. Simplified 14.1%

   \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{A \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. associate-*r* N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{\left(A \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. cancel-sign-sub-inv N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. unpow2 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. metadata-eval N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   11. *-lowering-*.f64 14.1

       \[\leadsto \frac{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

8. Simplified 14.1%

   \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
9. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{-2 \cdot \sqrt{\left(A \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}}}} \]

10. Applied egg-rr 14.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}{-2 \cdot \sqrt{A \cdot \left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right)}}}} \]

11. Taylor expanded in C around 0

    \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F} \cdot 1}{B}} \]

    2. *-rgt-identity N/A

       \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{A \cdot F}}{B}} \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{A \cdot F}}}{B} \]

    6. *-lowering-*.f64 3.9

       \[\leadsto -2 \cdot \frac{\sqrt{\color{blue}{A \cdot F}}}{B} \]

13. Simplified 3.9%

    \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{A \cdot F}}{B}} \]
14. Add Preprocessing

Alternative 16: 2.0% accurate, 18.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B}} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (sqrt (* F (/ 2.0 B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((F * (2.0 / B)));
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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
    code = sqrt((f * (2.0d0 / b)))
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt((F * (2.0 / B)));
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((F * (2.0 / B)))

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return sqrt(Float64(F * Float64(2.0 / B)))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt((F * (2.0 / B)));
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[Sqrt[N[(F * N[(2.0 / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 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. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. neg-lowering-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. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \sqrt{2}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 2.0

       \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \color{blue}{\sqrt{2}}\right) \]

5. Simplified 2.0%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right)}\right)\right) \]

   2. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

   3. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B} \cdot 2}} \]

   6. /-lowering-/.f64 2.0

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}} \cdot 2} \]

7. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   2. associate-/l* N/A

      \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

   4. /-lowering-/.f64 2.0

      \[\leadsto \sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 2.0%

   \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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))))