ABCF->ab-angle a

Percentage Accurate: 19.5% → 59.0%

Time: 20.7s

Alternatives: 21

Speedup: 16.4×

• could not determine a ground truth

  1. A = -6.220805173741362e+65
  2. B = -5.738260805902138e-76
  3. C = 8.97812798168438e-216
  4. F = 4.350858839923357e+104

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)} \cdot \sqrt{C}\right)}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \mathsf{fma}\left(\left(A + C\right) \cdot \left(A - C\right), \frac{-1}{C - A}, \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4
         (/
          (*
           2.0
           (* (sqrt (* F (fma C (* A -4.0) (fma B_m B_m 0.0)))) (sqrt C)))
          t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-171)
       (/
        (sqrt
         (*
          t_1
          (fma
           (* (+ A C) (- A C))
           (/ -1.0 (- C A))
           (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        t_2)
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt
            (*
             (fma 2.0 C (/ (* (* B_m B_m) -0.5) A))
             (* 2.0 (fma B_m B_m (* -4.0 (* A C))))))
           (sqrt F))
          t_2)
         (if (<= t_3 INFINITY)
           t_4
           (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = (2.0 * (sqrt((F * fma(C, (A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-171) {
		tmp = sqrt((t_1 * fma(((A + C) * (A - C)), (-1.0 / (C - A)), sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_2;
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(2.0, C, (((B_m * B_m) * -0.5) / A)) * (2.0 * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt(F)) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(Float64(2.0 * Float64(sqrt(Float64(F * fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(t_1 * fma(Float64(Float64(A + C) * Float64(A - C)), Float64(-1.0 / Float64(C - A)), sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / t_2);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)) * Float64(2.0 * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(F)) / t_2);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * N[(N[Sqrt[N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(A + C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(C - A), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.4%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 27.0

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

   5. Simplified 27.0%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 25.9

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

   8. Simplified 25.9%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

   10. Applied egg-rr 37.9%

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

   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))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 29.1

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

   5. Simplified 29.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

   7. Applied egg-rr 41.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 17.8

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 17.9

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 26.3

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\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}{C - A}, \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 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)} \cdot \sqrt{C}\right)}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4
         (/
          (*
           2.0
           (* (sqrt (* F (fma C (* A -4.0) (fma B_m B_m 0.0)))) (sqrt C)))
          t_2)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-171)
       (*
        (sqrt
         (*
          (* 2.0 (* F t_0))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_0))
       (if (<= t_3 0.0)
         (/
          (*
           (sqrt
            (*
             (fma 2.0 C (/ (* (* B_m B_m) -0.5) A))
             (* 2.0 (fma B_m B_m (* -4.0 (* A C))))))
           (sqrt F))
          t_2)
         (if (<= t_3 INFINITY)
           t_4
           (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = (2.0 * (sqrt((F * fma(C, (A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-171) {
		tmp = sqrt(((2.0 * (F * t_0)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_0);
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((fma(2.0, C, (((B_m * B_m) * -0.5) / A)) * (2.0 * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt(F)) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(Float64(2.0 * Float64(sqrt(Float64(F * fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_0));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A)) * Float64(2.0 * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(F)) / t_2);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(2.0 * N[(N[Sqrt[N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.4%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 27.0

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

   5. Simplified 27.0%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 25.9

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

   8. Simplified 25.9%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

   10. Applied egg-rr 37.9%

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

   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))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 29.1

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

   5. Simplified 29.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

   7. Applied egg-rr 41.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 17.8

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 17.9

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 26.3

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right) \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{F}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ t_5 := \frac{2 \cdot \left(\sqrt{F \cdot t\_1} \cdot \sqrt{C}\right)}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F} \cdot \sqrt{C \cdot t\_1}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (fma C (* A -4.0) (fma B_m B_m 0.0)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3))
        (t_5 (/ (* 2.0 (* (sqrt (* F t_1)) (sqrt C))) t_3)))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-171)
       (*
        (sqrt
         (*
          (* 2.0 (* F t_0))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_0))
       (if (<= t_4 0.0)
         (/ (* 2.0 (* (sqrt F) (sqrt (* C t_1)))) t_3)
         (if (<= t_4 INFINITY)
           t_5
           (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = fma(C, (A * -4.0), fma(B_m, B_m, 0.0));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = (2.0 * (sqrt((F * t_1)) * sqrt(C))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-171) {
		tmp = sqrt(((2.0 * (F * t_0)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_0);
	} else if (t_4 <= 0.0) {
		tmp = (2.0 * (sqrt(F) * sqrt((C * t_1)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	t_5 = Float64(Float64(2.0 * Float64(sqrt(Float64(F * t_1)) * sqrt(C))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_0));
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(sqrt(F) * sqrt(Float64(C * t_1)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-171], N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[(2.0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$5, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 9.4%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 27.0

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

   5. Simplified 27.0%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 25.9

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

   8. Simplified 25.9%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

   10. Applied egg-rr 37.9%

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

   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))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.7%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 25.5

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

   5. Simplified 25.5%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 25.6

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

   8. Simplified 25.6%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

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

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

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

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. +-lft-identity N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(C, -4 \cdot A, B \cdot \color{blue}{\left(0 + B\right)}\right) \cdot C} \cdot {F}^{\frac{1}{2}}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(C, -4 \cdot A, B \cdot \color{blue}{\left(B + 0\right)}\right) \cdot C} \cdot {F}^{\frac{1}{2}}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. distribute-rgt-out N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(C, -4 \cdot A, \color{blue}{B \cdot B + 0 \cdot B}\right) \cdot C} \cdot {F}^{\frac{1}{2}}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      16. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(C, -4 \cdot A, B \cdot B + \color{blue}{0}\right) \cdot C} \cdot {F}^{\frac{1}{2}}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(C, -4 \cdot A, \color{blue}{\mathsf{fma}\left(B, B, 0\right)}\right) \cdot C} \cdot {F}^{\frac{1}{2}}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   10. Applied egg-rr 34.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 17.8

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 17.9

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 26.3

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F} \cdot \sqrt{C \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)} \cdot \sqrt{C}\right)}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_4\right) \cdot \left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B\_m \cdot B\_m\right) \cdot -0.5}{A}\right)\right)}}{0 - t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (fma B_m B_m (* -4.0 (* A C)))))
   (if (<= t_3 (- INFINITY))
     (/
      (* 2.0 (* (sqrt (* F (fma C (* A -4.0) (fma B_m B_m 0.0)))) (sqrt C)))
      t_2)
     (if (<= t_3 -1e-171)
       (*
        (sqrt
         (*
          (* 2.0 (* F t_0))
          (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (/ -1.0 t_0))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* F t_4) (* 2.0 (fma 2.0 C (/ (* (* B_m B_m) -0.5) A)))))
          (- 0.0 t_4))
         (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = fma(B_m, B_m, (-4.0 * (A * C)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (2.0 * (sqrt((F * fma(C, (A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2;
	} else if (t_3 <= -1e-171) {
		tmp = sqrt(((2.0 * (F * t_0)) * ((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) * (-1.0 / t_0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_4) * (2.0 * fma(2.0, C, (((B_m * B_m) * -0.5) / A))))) / (0.0 - t_4);
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * Float64(sqrt(Float64(F * fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0)))) * sqrt(C))) / t_2);
	elseif (t_3 <= -1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) * Float64(-1.0 / t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_4) * Float64(2.0 * fma(2.0, C, Float64(Float64(Float64(B_m * B_m) * -0.5) / A))))) / Float64(0.0 - t_4));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(2.0 * N[(N[Sqrt[N[(F * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -1e-171], N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$4), $MachinePrecision] * N[(2.0 * N[(2.0 * C + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * -0.5), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 24.7

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

   5. Simplified 24.7%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 23.2

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

   8. Simplified 23.2%

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

      1. pow1/2 N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

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

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

   10. Applied egg-rr 38.4%

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

   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))) < -9.9999999999999998e-172

   1. Initial program 99.4%

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

      1. frac-2neg N/A

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

      2. div-inv N/A

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

      3. remove-double-neg N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. remove-double-neg N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if -9.9999999999999998e-172 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 12.4%

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

   3. Taylor expanded in A around -inf

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

      1. +-commutative N/A

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 30.5

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

   5. Simplified 30.5%

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

      1. distribute-frac-neg N/A

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

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

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

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

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

   7. Applied egg-rr 30.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 17.8

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 17.8%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 17.9

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 26.3

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

   9. Applied egg-rr 26.3%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{2 \cdot \left(\sqrt{F \cdot \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)} \cdot \sqrt{C}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)}}{0 - \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \mathsf{fma}\left(C, C, 0\right)\right)}\right)}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-32)
     (* -2.0 (/ (sqrt (* C (* F t_0))) t_0))
     (if (<= (pow B_m 2.0) 2e+283)
       (/
        (* (sqrt F) (sqrt (* 2.0 (+ C (sqrt (fma B_m B_m (fma C C 0.0)))))))
        (- 0.0 B_m))
       (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = -2.0 * (sqrt((C * (F * t_0))) / t_0);
	} else if (pow(B_m, 2.0) <= 2e+283) {
		tmp = (sqrt(F) * sqrt((2.0 * (C + sqrt(fma(B_m, B_m, fma(C, C, 0.0))))))) / (0.0 - B_m);
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * Float64(F * t_0))) / t_0));
	elseif ((B_m ^ 2.0) <= 2e+283)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(2.0 * Float64(C + sqrt(fma(B_m, B_m, fma(C, C, 0.0))))))) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(-2.0 * N[(N[Sqrt[N[(C * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+283], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 29.6

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

   5. Simplified 29.6%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 29.7

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

   8. Simplified 29.7%

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

   9. Taylor expanded in F around 0

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

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

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

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

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

   11. Simplified 29.6%

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

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999991e283

   1. Initial program 31.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

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

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

   5. Simplified 16.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 18.0

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 18.0%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

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

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

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

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

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

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

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

         \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}\right)} \cdot \sqrt{F}}{0 - B} \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(C + 0\right)}\right)}\right)} \cdot \sqrt{F}}{0 - B} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C + 0 \cdot C}\right)}\right)} \cdot \sqrt{F}}{0 - B} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C + \color{blue}{0}\right)}\right)} \cdot \sqrt{F}}{0 - B} \]

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

          \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{\mathsf{fma}\left(C, C, 0\right)}\right)}\right)} \cdot \sqrt{F}}{0 - B} \]

      13. sqrt-lowering-sqrt.f64 18.0

          \[\leadsto \frac{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}\right)} \cdot \color{blue}{\sqrt{F}}}{0 - B} \]

   9. Applied egg-rr 18.0%

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}\right)} \cdot \sqrt{F}}}{0 - B} \]

   if 1.99999999999999991e283 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.1%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 29.5

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 29.5%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 29.7

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 44.4

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

   9. Applied egg-rr 44.4%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+283}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}\right)}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (* -2.0 (sqrt (/ (* C F) (fma B_m B_m (* -4.0 (* A C))))))
   (if (<= (pow B_m 2.0) 1e+262)
     (/ (sqrt (* (* 2.0 F) (+ C (sqrt (fma B_m B_m (* C C)))))) (- 0.0 B_m))
     (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = -2.0 * sqrt(((C * F) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else if (pow(B_m, 2.0) <= 1e+262) {
		tmp = sqrt(((2.0 * F) * (C + sqrt(fma(B_m, B_m, (C * C)))))) / (0.0 - B_m);
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	elseif ((B_m ^ 2.0) <= 1e+262)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+262], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 29.6

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

   5. Simplified 29.6%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. *-lowering-*.f64 21.8

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

   8. Simplified 21.8%

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

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64)) < 1e262

   1. Initial program 34.1%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

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

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

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-frac-neg2 N/A

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

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

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

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

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

   7. Applied egg-rr 17.7%

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

   if 1e262 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

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

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

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

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

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

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 28.9

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 28.9%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. sqrt-unprod N/A

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

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

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

      5. rem-square-sqrt N/A

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

      6. sqrt-unprod N/A

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

      7. sqrt-unprod N/A

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

      8. +-lft-identity N/A

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

      9. +-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. sqrt-unprod N/A

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

      12. sqrt-unprod N/A

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

      13. rem-square-sqrt N/A

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

      14. mul0-lft N/A

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

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

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 29.1

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 29.1%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. sqrt-undiv N/A

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

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

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 42.7

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

   9. Applied egg-rr 42.7%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+262}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m + C}\right)}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-32)
     (* -2.0 (/ (sqrt (* C (* F t_0))) t_0))
     (/ (* (sqrt 2.0) (* (sqrt F) (sqrt (+ B_m C)))) (- 0.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = -2.0 * (sqrt((C * (F * t_0))) / t_0);
	} else {
		tmp = (sqrt(2.0) * (sqrt(F) * sqrt((B_m + C)))) / (0.0 - B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * Float64(F * t_0))) / t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(F) * sqrt(Float64(B_m + C)))) / Float64(0.0 - B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(-2.0 * N[(N[Sqrt[N[(C * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 29.6

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

   5. Simplified 29.6%

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

   6. Taylor expanded in F around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. *-lowering-*.f64 29.7

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

   8. Simplified 29.7%

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

   9. Taylor expanded in F around 0

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

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

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

       2. associate-*r/ N/A

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

       3. *-rgt-identity N/A

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

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

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

       5. metadata-eval N/A

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

       6. +-commutative N/A

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

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

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

   11. Simplified 29.6%

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

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

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

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

   5. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

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

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 10.0

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 10.0%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
   9. Step-by-step derivation

   10. Simplified 29.9%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{2 \cdot \sqrt{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)}}{-4 \cdot \left(0 - A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m + C}\right)}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (/
    (* 2.0 (sqrt (* (fma B_m B_m (* -4.0 (* A C))) (* C F))))
    (* -4.0 (- 0.0 (* A C))))
   (/ (* (sqrt 2.0) (* (sqrt F) (sqrt (+ B_m C)))) (- 0.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = (2.0 * sqrt((fma(B_m, B_m, (-4.0 * (A * C))) * (C * F)))) / (-4.0 * (0.0 - (A * C)));
	} else {
		tmp = (sqrt(2.0) * (sqrt(F) * sqrt((B_m + C)))) / (0.0 - B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(Float64(2.0 * sqrt(Float64(fma(B_m, B_m, Float64(-4.0 * Float64(A * C))) * Float64(C * F)))) / Float64(-4.0 * Float64(0.0 - Float64(A * C))));
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(F) * sqrt(Float64(B_m + C)))) / Float64(0.0 - B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(N[(2.0 * N[Sqrt[N[(N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(0.0 - N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. mul0-lft N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft1-in N/A

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

      7. distribute-lft-in N/A

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

      8. distribute-rgt-in N/A

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

      9. distribute-lft1-in N/A

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

      10. metadata-eval N/A

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

      11. mul0-lft N/A

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

      12. mul0-lft N/A

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

      13. accelerator-lowering-fma.f64 29.6

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

   5. Simplified 29.6%

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

   6. Taylor expanded in F around 0

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

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

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

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

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 29.7

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 29.7%

      \[\leadsto \frac{-\color{blue}{2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Taylor expanded in B around 0

      \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)}{-4 \cdot \color{blue}{\left(C \cdot A\right)}} \]

       3. *-lowering-*.f64 28.6

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{-4 \cdot \color{blue}{\left(C \cdot A\right)}} \]

   11. Simplified 28.6%

       \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}{\color{blue}{-4 \cdot \left(C \cdot A\right)}} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left(C + \sqrt{C \cdot C + B \cdot B}\right) \cdot F}}}{0 - B} \]

      2. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)}}{0 - B} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 10.0

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 10.0%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
   9. Step-by-step derivation

   10. Simplified 29.9%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;\frac{2 \cdot \sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)}}{-4 \cdot \left(0 - A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-127}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B\_m + C}\right)}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-127)
   (* -2.0 (sqrt (/ (* C F) (fma B_m B_m (* -4.0 (* A C))))))
   (/ (* (sqrt 2.0) (* (sqrt F) (sqrt (+ B_m C)))) (- 0.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-127) {
		tmp = -2.0 * sqrt(((C * F) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else {
		tmp = (sqrt(2.0) * (sqrt(F) * sqrt((B_m + C)))) / (0.0 - B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-127)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(F) * sqrt(Float64(B_m + C)))) / Float64(0.0 - B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-127], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-127

   1. Initial program 14.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot \color{blue}{\left(2 + 0\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2 + C \cdot 0\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(0 \cdot \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{A}{C}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{A}{C}\right)} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \left(\color{blue}{0} \cdot \frac{A}{C}\right) \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. accelerator-lowering-fma.f64 30.2

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 30.2%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{C \cdot F}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}} \]

      6. unpow2 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      7. metadata-eval N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}} \]

      10. *-lowering-*.f64 22.9

          \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}} \]

   8. Simplified 22.9%

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

   if 2.0000000000000001e-127 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 10.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left(C + \sqrt{C \cdot C + B \cdot B}\right) \cdot F}}}{0 - B} \]

      2. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)}}{0 - B} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 11.1

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 11.1%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
   9. Step-by-step derivation

   10. Simplified 28.9%

       \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{B}} \cdot \sqrt{F}\right)}{0 - B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-127}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-32)
   (* -2.0 (sqrt (/ (* C F) (fma B_m B_m (* -4.0 (* A C))))))
   (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-32) {
		tmp = -2.0 * sqrt(((C * F) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-32)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(C * F) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-32], N[(-2.0 * N[Sqrt[N[(N[(C * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000006e-32

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot \color{blue}{\left(2 + 0\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2 + C \cdot 0\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(0 \cdot \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{A}{C}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{A}{C}\right)} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \left(\color{blue}{0} \cdot \frac{A}{C}\right) \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. accelerator-lowering-fma.f64 29.6

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 29.6%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\sqrt{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\color{blue}{\frac{C \cdot F}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{\color{blue}{C \cdot F}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}}} \]

      6. unpow2 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      7. metadata-eval N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}} \]

      10. *-lowering-*.f64 21.8

          \[\leadsto -2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}} \]

   8. Simplified 21.8%

      \[\leadsto \color{blue}{-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]

   if 1.00000000000000006e-32 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 22.1

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 22.1%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 22.3

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 22.3%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 30.3

         \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   9. Applied egg-rr 30.3%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-32}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)\\ \mathbf{if}\;B\_m \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(C \cdot F\right)} \cdot -2}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \mathsf{fma}\left(C, C, 0\right)\right)}\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (fma B_m B_m 0.0))))
   (if (<= B_m 5.2e-14)
     (/ (* (sqrt (* t_0 (* C F))) -2.0) t_0)
     (if (<= B_m 1.7e+138)
       (*
        (sqrt (* 2.0 (+ C (sqrt (fma B_m B_m (fma C C 0.0))))))
        (* (sqrt F) (/ -1.0 B_m)))
       (* (sqrt (/ 2.0 B_m)) (- 0.0 (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), fma(B_m, B_m, 0.0));
	double tmp;
	if (B_m <= 5.2e-14) {
		tmp = (sqrt((t_0 * (C * F))) * -2.0) / t_0;
	} else if (B_m <= 1.7e+138) {
		tmp = sqrt((2.0 * (C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) * (sqrt(F) * (-1.0 / B_m));
	} else {
		tmp = sqrt((2.0 / B_m)) * (0.0 - sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0))
	tmp = 0.0
	if (B_m <= 5.2e-14)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(C * F))) * -2.0) / t_0);
	elseif (B_m <= 1.7e+138)
		tmp = Float64(sqrt(Float64(2.0 * Float64(C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) * Float64(sqrt(F) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(0.0 - sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5.2e-14], N[(N[(N[Sqrt[N[(t$95$0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.7e+138], N[(N[Sqrt[N[(2.0 * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.19999999999999993e-14

   1. Initial program 15.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot \color{blue}{\left(2 + 0\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2 + C \cdot 0\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(0 \cdot \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{A}{C}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{A}{C}\right)} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \left(\color{blue}{0} \cdot \frac{A}{C}\right) \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. accelerator-lowering-fma.f64 21.7

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 21.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 21.7

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 21.7%

      \[\leadsto \frac{-\color{blue}{2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

   10. Applied egg-rr 21.7%

       \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(C, -4 \cdot A, \mathsf{fma}\left(B, B, 0\right)\right) \cdot \left(F \cdot C\right)} \cdot -2}{\mathsf{fma}\left(C, -4 \cdot A, \mathsf{fma}\left(B, B, 0\right)\right)}} \]

   if 5.19999999999999993e-14 < B < 1.70000000000000006e138

   1. Initial program 37.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left(C + \sqrt{C \cdot C + B \cdot B}\right) \cdot F}}}{0 - B} \]

      2. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)}}{0 - B} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 34.1

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 34.1%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]
   8. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \sqrt{F}\right)\right) \cdot \frac{1}{0 - B}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{C + \sqrt{B \cdot B + C \cdot C}}\right) \cdot \sqrt{F}\right)} \cdot \frac{1}{0 - B} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{C + \sqrt{B \cdot B + C \cdot C}}\right) \cdot \left(\sqrt{F} \cdot \frac{1}{0 - B}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{C + \sqrt{B \cdot B + C \cdot C}}\right) \cdot \left(\sqrt{F} \cdot \frac{1}{0 - B}\right)} \]

   9. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)} \]

   if 1.70000000000000006e138 < B

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 54.3

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 54.6

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left({F}^{\frac{1}{2}} \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-undiv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      13. /-lowering-/.f64 84.1

          \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 84.1%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right) \cdot \left(C \cdot F\right)} \cdot -2}{\mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B\_m, B\_m, 0\right)\right)\\ \mathbf{if}\;B\_m \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(C \cdot F\right)} \cdot -2}{t\_0}\\ \mathbf{elif}\;B\_m \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \mathsf{fma}\left(C, C, 0\right)\right)}}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (fma B_m B_m 0.0))))
   (if (<= B_m 2.4e-14)
     (/ (* (sqrt (* t_0 (* C F))) -2.0) t_0)
     (if (<= B_m 2.1e+134)
       (/
        (* (sqrt (* 2.0 F)) (sqrt (+ C (sqrt (fma B_m B_m (fma C C 0.0))))))
        (- 0.0 B_m))
       (* (sqrt (/ 2.0 B_m)) (- 0.0 (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(C, (A * -4.0), fma(B_m, B_m, 0.0));
	double tmp;
	if (B_m <= 2.4e-14) {
		tmp = (sqrt((t_0 * (C * F))) * -2.0) / t_0;
	} else if (B_m <= 2.1e+134) {
		tmp = (sqrt((2.0 * F)) * sqrt((C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) / (0.0 - B_m);
	} else {
		tmp = sqrt((2.0 / B_m)) * (0.0 - sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(C, Float64(A * -4.0), fma(B_m, B_m, 0.0))
	tmp = 0.0
	if (B_m <= 2.4e-14)
		tmp = Float64(Float64(sqrt(Float64(t_0 * Float64(C * F))) * -2.0) / t_0);
	elseif (B_m <= 2.1e+134)
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(0.0 - sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.4e-14], N[(N[(N[Sqrt[N[(t$95$0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.1e+134], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.4e-14

   1. Initial program 15.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot \color{blue}{\left(2 + 0\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2 + C \cdot 0\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(0 \cdot \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{A}{C}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{A}{C}\right)} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \left(\color{blue}{0} \cdot \frac{A}{C}\right) \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. accelerator-lowering-fma.f64 21.7

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 21.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 21.7

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 21.7%

      \[\leadsto \frac{-\color{blue}{2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{\color{blue}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}} \]

   10. Applied egg-rr 21.7%

       \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(C, -4 \cdot A, \mathsf{fma}\left(B, B, 0\right)\right) \cdot \left(F \cdot C\right)} \cdot -2}{\mathsf{fma}\left(C, -4 \cdot A, \mathsf{fma}\left(B, B, 0\right)\right)}} \]

   if 2.4e-14 < B < 2.1000000000000001e134

   1. Initial program 37.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left(C + \sqrt{C \cdot C + B \cdot B}\right) \cdot F}}}{0 - B} \]

      2. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)}}{0 - B} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 34.1

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 34.1%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \sqrt{F}\right) \cdot \sqrt{2}}}{0 - B} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \left(\sqrt{F} \cdot \sqrt{2}\right)}}{0 - B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \color{blue}{\sqrt{F \cdot 2}}}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \sqrt{F \cdot 2}}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{C + \sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(C + 0\right)}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C + 0 \cdot C}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C + \color{blue}{0}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{\mathsf{fma}\left(C, C, 0\right)}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \color{blue}{\sqrt{F \cdot 2}}}{0 - B} \]

      14. *-lowering-*.f64 34.3

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \sqrt{\color{blue}{F \cdot 2}}}{0 - B} \]

   9. Applied egg-rr 34.3%

      \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \sqrt{F \cdot 2}}}{0 - B} \]

   if 2.1000000000000001e134 < B

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 54.3

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 54.6

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left({F}^{\frac{1}{2}} \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-undiv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      13. /-lowering-/.f64 84.1

          \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 84.1%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right) \cdot \left(C \cdot F\right)} \cdot -2}{\mathsf{fma}\left(C, A \cdot -4, \mathsf{fma}\left(B, B, 0\right)\right)}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 3.3 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 9 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \mathsf{fma}\left(C, C, 0\right)\right)}}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B_m B_m))))
   (if (<= B_m 3.3e-14)
     (* -2.0 (/ (sqrt (* C (* F t_0))) t_0))
     (if (<= B_m 9e+133)
       (/
        (* (sqrt (* 2.0 F)) (sqrt (+ C (sqrt (fma B_m B_m (fma C C 0.0))))))
        (- 0.0 B_m))
       (* (sqrt (/ 2.0 B_m)) (- 0.0 (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B_m * B_m));
	double tmp;
	if (B_m <= 3.3e-14) {
		tmp = -2.0 * (sqrt((C * (F * t_0))) / t_0);
	} else if (B_m <= 9e+133) {
		tmp = (sqrt((2.0 * F)) * sqrt((C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) / (0.0 - B_m);
	} else {
		tmp = sqrt((2.0 / B_m)) * (0.0 - sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 3.3e-14)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * Float64(F * t_0))) / t_0));
	elseif (B_m <= 9e+133)
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) * sqrt(Float64(C + sqrt(fma(B_m, B_m, fma(C, C, 0.0)))))) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(0.0 - sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.3e-14], N[(-2.0 * N[(N[Sqrt[N[(C * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 9e+133], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C + 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.2999999999999998e-14

   1. Initial program 15.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around -inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot \color{blue}{\left(2 + 0\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2 + C \cdot 0\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(0 \cdot \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot \frac{A}{C}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C \cdot 2 + C \cdot \color{blue}{\left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. distribute-rgt-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(2 \cdot C + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right) \cdot C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{\left(\left(-1 + 1\right) \cdot \frac{A}{C}\right)} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \left(\color{blue}{0} \cdot \frac{A}{C}\right) \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0} \cdot C\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. mul0-lft N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C + \color{blue}{0}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. accelerator-lowering-fma.f64 21.7

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 21.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, 0\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2 \cdot \sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\color{blue}{\left(C \cdot F\right)} \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\left({B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \left(B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. *-lowering-*.f64 21.7

          \[\leadsto \frac{-2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   8. Simplified 21.7%

      \[\leadsto \frac{-\color{blue}{2 \cdot \sqrt{\left(C \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{1}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot \frac{1}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\right)} \]

       2. associate-*r/ N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)} \cdot 1}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]

       3. *-rgt-identity N/A

          \[\leadsto -2 \cdot \frac{\color{blue}{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

       4. cancel-sign-sub-inv N/A

          \[\leadsto -2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

       5. metadata-eval N/A

          \[\leadsto -2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}{{B}^{2} + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

       6. +-commutative N/A

          \[\leadsto -2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto -2 \cdot \color{blue}{\frac{\sqrt{C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot C\right) + {B}^{2}\right)\right)}}{-4 \cdot \left(A \cdot C\right) + {B}^{2}}} \]

   11. Simplified 21.7%

       \[\leadsto \color{blue}{-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   if 3.2999999999999998e-14 < B < 8.9999999999999997e133

   1. Initial program 37.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\left(C + \sqrt{C \cdot C + B \cdot B}\right) \cdot F}}}{0 - B} \]

      2. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \sqrt{F}\right)}}{0 - B} \]

      3. pow1/2 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot \color{blue}{{F}^{\frac{1}{2}}}\right)}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{C \cdot C + B \cdot B}} \cdot {F}^{\frac{1}{2}}\right)}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\color{blue}{\sqrt{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{C + \sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \color{blue}{\sqrt{C \cdot C + B \cdot B}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{B \cdot B + C \cdot C}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C}\right)}} \cdot {F}^{\frac{1}{2}}\right)}{0 - B} \]

      11. pow1/2 N/A

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

      12. sqrt-lowering-sqrt.f64 34.1

          \[\leadsto \frac{\sqrt{2} \cdot \left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \color{blue}{\sqrt{F}}\right)}{0 - B} \]

   7. Applied egg-rr 34.1%

      \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\left(\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}} \cdot \sqrt{F}\right)}}{0 - B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \sqrt{F}\right) \cdot \sqrt{2}}}{0 - B} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \left(\sqrt{F} \cdot \sqrt{2}\right)}}{0 - B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \color{blue}{\sqrt{F \cdot 2}}}{0 - B} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}} \cdot \sqrt{F \cdot 2}}}{0 - B} \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{C + \sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{\color{blue}{\mathsf{fma}\left(B, B, C \cdot C\right)}}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(C + 0\right)}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{C \cdot C + 0 \cdot C}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      11. mul0-lft N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, C \cdot C + \color{blue}{0}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \color{blue}{\mathsf{fma}\left(C, C, 0\right)}\right)}} \cdot \sqrt{F \cdot 2}}{0 - B} \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \color{blue}{\sqrt{F \cdot 2}}}{0 - B} \]

      14. *-lowering-*.f64 34.3

          \[\leadsto \frac{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \sqrt{\color{blue}{F \cdot 2}}}{0 - B} \]

   9. Applied egg-rr 34.3%

      \[\leadsto \frac{\color{blue}{\sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}} \cdot \sqrt{F \cdot 2}}}{0 - B} \]

   if 8.9999999999999997e133 < B

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 54.3

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 54.6

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left({F}^{\frac{1}{2}} \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-undiv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      13. /-lowering-/.f64 84.1

          \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 84.1%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.3 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot \left(F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+133}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B, B, \mathsf{fma}\left(C, C, 0\right)\right)}}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(0 - \sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 35.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 5.8e+192)
   (/ (sqrt (* 2.0 F)) (- 0.0 (sqrt B_m)))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5.8e+192) {
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 5.8d+192) then
        tmp = sqrt((2.0d0 * f)) / (0.0d0 - sqrt(b_m))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 5.8e+192) {
		tmp = Math.sqrt((2.0 * F)) / (0.0 - Math.sqrt(B_m));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 5.8e+192:
		tmp = math.sqrt((2.0 * F)) / (0.0 - math.sqrt(B_m))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 5.8e+192)
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(0.0 - sqrt(B_m)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 5.8e+192)
		tmp = sqrt((2.0 * F)) / (0.0 - sqrt(B_m));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 5.8e+192], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(0.0 - N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.8000000000000003e192

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 15.2

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.2%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 15.3

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 15.3%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 20.1

         \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   9. Applied egg-rr 20.1%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]

   if 5.8000000000000003e192 < C

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}{B}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}\right)}{B}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{C \cdot F}\right)}{B} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2} \cdot \sqrt{C \cdot F}\right)}{B} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sqrt{C \cdot F}}}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot \sqrt{C \cdot F}}{B} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{C \cdot F}}{B}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot F}}}{B} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{C \cdot F}}}{B} \]

      13. *-commutative N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

      14. *-lowering-*.f64 8.8

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

   8. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.8 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{0 - \sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;0 - \frac{\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 4.5e+192)
   (- 0.0 (/ (sqrt F) (sqrt (* B_m 0.5))))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.5e+192) {
		tmp = 0.0 - (sqrt(F) / sqrt((B_m * 0.5)));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 4.5d+192) then
        tmp = 0.0d0 - (sqrt(f) / sqrt((b_m * 0.5d0)))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.5e+192) {
		tmp = 0.0 - (Math.sqrt(F) / Math.sqrt((B_m * 0.5)));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 4.5e+192:
		tmp = 0.0 - (math.sqrt(F) / math.sqrt((B_m * 0.5)))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 4.5e+192)
		tmp = Float64(0.0 - Float64(sqrt(F) / sqrt(Float64(B_m * 0.5))));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 4.5e+192)
		tmp = 0.0 - (sqrt(F) / sqrt((B_m * 0.5)));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 4.5e+192], N[(0.0 - N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.5e192

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 15.2

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.2%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 15.3

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 15.3%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F \cdot 2}}}{\sqrt{B}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\color{blue}{F \cdot 2}}}{\sqrt{B}}\right) \]

      8. sqrt-lowering-sqrt.f64 20.1

         \[\leadsto -\frac{\sqrt{F \cdot 2}}{\color{blue}{\sqrt{B}}} \]

   9. Applied egg-rr 20.1%

      \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot 2}}{\sqrt{B}}} \]
   10. Step-by-step derivation

       1. sqrt-undiv N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F \cdot 2}{B}}}\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F \cdot \color{blue}{\frac{1}{\frac{B}{2}}}}\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F}{\frac{B}{2}}}}\right) \]

       5. sqrt-div N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{\frac{B}{2}}}}\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{F}}{\sqrt{\frac{B}{2}}}}\right) \]

       7. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{F}}}{\sqrt{\frac{B}{2}}}\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{F}}{\color{blue}{\sqrt{\frac{B}{2}}}}\right) \]

       9. div-inv N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{F}}{\sqrt{\color{blue}{B \cdot \frac{1}{2}}}}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{neg}\left(\frac{\sqrt{F}}{\sqrt{B \cdot \color{blue}{\frac{1}{2}}}}\right) \]

       11. *-lowering-*.f64 20.1

           \[\leadsto -\frac{\sqrt{F}}{\sqrt{\color{blue}{B \cdot 0.5}}} \]

   11. Applied egg-rr 20.1%

       \[\leadsto -\color{blue}{\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}} \]

   if 4.5e192 < C

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}{B}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}\right)}{B}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{C \cdot F}\right)}{B} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2} \cdot \sqrt{C \cdot F}\right)}{B} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sqrt{C \cdot F}}}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot \sqrt{C \cdot F}}{B} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{C \cdot F}}{B}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot F}}}{B} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{C \cdot F}}}{B} \]

      13. *-commutative N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

      14. *-lowering-*.f64 8.8

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

   8. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;0 - \frac{\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 7.6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{\frac{2}{B\_m}} \cdot \left(0 - \sqrt{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 7.6e+192)
   (* (sqrt (/ 2.0 B_m)) (- 0.0 (sqrt F)))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 7.6e+192) {
		tmp = sqrt((2.0 / B_m)) * (0.0 - sqrt(F));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 7.6d+192) then
        tmp = sqrt((2.0d0 / b_m)) * (0.0d0 - sqrt(f))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 7.6e+192) {
		tmp = Math.sqrt((2.0 / B_m)) * (0.0 - Math.sqrt(F));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 7.6e+192:
		tmp = math.sqrt((2.0 / B_m)) * (0.0 - math.sqrt(F))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 7.6e+192)
		tmp = Float64(sqrt(Float64(2.0 / B_m)) * Float64(0.0 - sqrt(F)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 7.6e+192)
		tmp = sqrt((2.0 / B_m)) * (0.0 - sqrt(F));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 7.6e+192], N[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 7.5999999999999999e192

   1. Initial program 18.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 15.2

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.2%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 15.3

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 15.3%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

      5. sqrt-prod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}}\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}}} \cdot \sqrt{\frac{2}{B}}\right) \]

      7. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left({F}^{\frac{1}{2}} \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{{F}^{\frac{1}{2}} \cdot \frac{\sqrt{2}}{\sqrt{B}}}\right) \]

      9. pow1/2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{2}}{\sqrt{B}}\right) \]

      11. sqrt-undiv N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{F} \cdot \color{blue}{\sqrt{\frac{2}{B}}}\right) \]

      13. /-lowering-/.f64 20.0

          \[\leadsto -\sqrt{F} \cdot \sqrt{\color{blue}{\frac{2}{B}}} \]

   9. Applied egg-rr 20.0%

      \[\leadsto -\color{blue}{\sqrt{F} \cdot \sqrt{\frac{2}{B}}} \]

   if 7.5999999999999999e192 < C

   1. Initial program 1.8%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}{B}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}\right)}{B}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{C \cdot F}\right)}{B} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2} \cdot \sqrt{C \cdot F}\right)}{B} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sqrt{C \cdot F}}}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot \sqrt{C \cdot F}}{B} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{C \cdot F}}{B}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot F}}}{B} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{C \cdot F}}}{B} \]

      13. *-commutative N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

      14. *-lowering-*.f64 8.8

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

   8. Simplified 8.8%

      \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 7.6 \cdot 10^{+192}:\\ \;\;\;\;\sqrt{\frac{2}{B}} \cdot \left(0 - \sqrt{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.7 \cdot 10^{+108}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot F}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 4.7e+108)
   (- 0.0 (sqrt (/ (* 2.0 F) B_m)))
   (/ (* -2.0 (sqrt (* C F))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.7e+108) {
		tmp = 0.0 - sqrt(((2.0 * F) / B_m));
	} else {
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 4.7d+108) then
        tmp = 0.0d0 - sqrt(((2.0d0 * f) / b_m))
    else
        tmp = ((-2.0d0) * sqrt((c * f))) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.7e+108) {
		tmp = 0.0 - Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = (-2.0 * Math.sqrt((C * F))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 4.7e+108:
		tmp = 0.0 - math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = (-2.0 * math.sqrt((C * F))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 4.7e+108)
		tmp = Float64(0.0 - sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(C * F))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 4.7e+108)
		tmp = 0.0 - sqrt(((2.0 * F) / B_m));
	else
		tmp = (-2.0 * sqrt((C * F))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 4.7e+108], N[(0.0 - N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.6999999999999996e108

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in B around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

      4. *-commutative N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

      8. /-lowering-/.f64 15.9

         \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

   5. Simplified 15.9%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

      5. rem-square-sqrt N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      8. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      12. sqrt-unprod N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      13. rem-square-sqrt N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

      14. mul0-lft N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

      16. /-lowering-/.f64 16.0

          \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

   7. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F \cdot 2}{B}}}\right) \]

      5. *-lowering-*.f64 16.0

         \[\leadsto -\sqrt{\frac{\color{blue}{F \cdot 2}}{B}} \]

   9. Applied egg-rr 16.0%

      \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

   if 4.6999999999999996e108 < C

   1. Initial program 6.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

   5. Simplified 3.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]

   6. Taylor expanded in C around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{{\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}{B}}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\sqrt{C \cdot F} \cdot {\left(\sqrt{2}\right)}^{2}\right)}{B}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{\left(\sqrt{2}\right)}^{2} \cdot \sqrt{C \cdot F}}\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)} \cdot \sqrt{C \cdot F}\right)}{B} \]

      7. rem-square-sqrt N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{2} \cdot \sqrt{C \cdot F}\right)}{B} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \sqrt{C \cdot F}}}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-2} \cdot \sqrt{C \cdot F}}{B} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{C \cdot F}}{B}} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 \cdot \sqrt{C \cdot F}}}{B} \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \frac{-2 \cdot \color{blue}{\sqrt{C \cdot F}}}{B} \]

      13. *-commutative N/A

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

      14. *-lowering-*.f64 8.5

          \[\leadsto \frac{-2 \cdot \sqrt{\color{blue}{F \cdot C}}}{B} \]

   8. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot C}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.7 \cdot 10^{+108}:\\ \;\;\;\;0 - \sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 27.1% accurate, 14.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ 0 - \sqrt{\left|\mathsf{fma}\left(F, \frac{2}{B\_m}, 0\right)\right|} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (- 0.0 (sqrt (fabs (fma F (/ 2.0 B_m) 0.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt(fabs(fma(F, (2.0 / B_m), 0.0)));
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(abs(fma(F, Float64(2.0 / B_m), 0.0))))
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[Abs[N[(F * N[(2.0 / B$95$m), $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. *-commutative N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. /-lowering-/.f64 13.8

      \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

   6. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

   7. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   8. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   12. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   14. mul0-lft N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

   16. /-lowering-/.f64 13.9

       \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

7. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   2. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{\left(2 \cdot \frac{F}{B}\right) \cdot \left(2 \cdot \frac{F}{B}\right)}}}\right) \]

   4. rem-sqrt-square N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left|2 \cdot \frac{F}{B}\right|}}\right) \]

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left|2 \cdot \frac{F}{B}\right|}}\right) \]

   6. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left|\color{blue}{2 \cdot \frac{F}{B} + 0}\right|}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left|\color{blue}{\frac{2 \cdot F}{B}} + 0\right|}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left|\frac{\color{blue}{F \cdot 2}}{B} + 0\right|}\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left|\color{blue}{F \cdot \frac{2}{B}} + 0\right|}\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\left|\color{blue}{\mathsf{fma}\left(F, \frac{2}{B}, 0\right)}\right|}\right) \]

   11. /-lowering-/.f64 24.7

       \[\leadsto -\sqrt{\left|\mathsf{fma}\left(F, \color{blue}{\frac{2}{B}}, 0\right)\right|} \]

9. Applied egg-rr 24.7%

   \[\leadsto -\sqrt{\color{blue}{\left|\mathsf{fma}\left(F, \frac{2}{B}, 0\right)\right|}} \]

10. Final simplification 24.7%

    \[\leadsto 0 - \sqrt{\left|\mathsf{fma}\left(F, \frac{2}{B}, 0\right)\right|} \]
11. Add Preprocessing

Alternative 19: 27.0% accurate, 16.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ 0 - \sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- 0.0 (sqrt (/ (* 2.0 F) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt(((2.0 * F) / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 0.0d0 - sqrt(((2.0d0 * f) / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.sqrt(((2.0 * F) / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return 0.0 - math.sqrt(((2.0 * F) / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(Float64(Float64(2.0 * F) / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - sqrt(((2.0 * F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. *-commutative N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. /-lowering-/.f64 13.8

      \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

   6. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

   7. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   8. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   12. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   14. mul0-lft N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

   16. /-lowering-/.f64 13.9

       \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

7. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{F \cdot 2}{B}}}\right) \]

   5. *-lowering-*.f64 13.9

      \[\leadsto -\sqrt{\frac{\color{blue}{F \cdot 2}}{B}} \]

9. Applied egg-rr 13.9%

   \[\leadsto -\sqrt{\color{blue}{\frac{F \cdot 2}{B}}} \]

10. Final simplification 13.9%

    \[\leadsto 0 - \sqrt{\frac{2 \cdot F}{B}} \]
11. Add Preprocessing

Alternative 20: 27.0% accurate, 16.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ 0 - \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- 0.0 (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 0.0d0 - sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return 0.0 - math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   3. --lowering--.f64 N/A

      \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   4. *-commutative N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{F}{B}} \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto 0 - \sqrt{2} \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. /-lowering-/.f64 13.8

      \[\leadsto 0 - \sqrt{2} \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Simplified 13.8%

   \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \]

   3. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2 \cdot \frac{F}{B}}}\right) \]

   5. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}} \cdot \sqrt{2 \cdot \frac{F}{B}}}}\right) \]

   6. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)} \cdot \sqrt{2 \cdot \frac{F}{B}}}\right) \]

   7. sqrt-unprod N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   8. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(0 + \sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}} + 0\right)}}\right) \]

   10. distribute-rgt-out N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}}\right) \]

   11. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   12. sqrt-unprod N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\sqrt{2 \cdot \frac{F}{B}} \cdot \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   13. rem-square-sqrt N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}} + 0 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right)}\right) \]

   14. mul0-lft N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{2 \cdot \frac{F}{B} + \color{blue}{0}}\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}}\right) \]

   16. /-lowering-/.f64 13.9

       \[\leadsto -\sqrt{\mathsf{fma}\left(2, \color{blue}{\frac{F}{B}}, 0\right)} \]

7. Applied egg-rr 13.9%

   \[\leadsto \color{blue}{-\sqrt{\mathsf{fma}\left(2, \frac{F}{B}, 0\right)}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{2 \cdot \frac{F}{B}}}\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{\frac{2 \cdot F}{B}}}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{\color{blue}{F \cdot 2}}{B}}\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\color{blue}{F \cdot \frac{2}{B}}}\right) \]

   6. /-lowering-/.f64 13.9

      \[\leadsto -\sqrt{F \cdot \color{blue}{\frac{2}{B}}} \]

9. Applied egg-rr 13.9%

   \[\leadsto -\sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]

10. Final simplification 13.9%

    \[\leadsto 0 - \sqrt{F \cdot \frac{2}{B}} \]
11. Add Preprocessing

Alternative 21: 3.8% accurate, 40.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{0}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ 0.0 B_m))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return 0.0 / B_m;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 0.0d0 / b_m
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return 0.0 / B_m;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return 0.0 / B_m

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(0.0 / B_m)
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = 0.0 / B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(0.0 / B$95$m), $MachinePrecision]

Derivation

1. Initial program 16.2%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{B}}\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\mathsf{neg}\left(B\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{\color{blue}{-1 \cdot B}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}{-1 \cdot B}} \]

5. Simplified 7.9%

   \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(C, C, B \cdot B\right)}\right)}}{0 - B}} \]

6. Taylor expanded in C around -inf

   \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{-1 \cdot C}\right)}}{0 - B} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{\left(\mathsf{neg}\left(C\right)\right)}\right)}}{0 - B} \]

   2. neg-sub0 N/A

      \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{\left(0 - C\right)}\right)}}{0 - B} \]

   3. --lowering--.f64 3.9

      \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{\left(0 - C\right)}\right)}}{0 - B} \]

8. Simplified 3.9%

   \[\leadsto \frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C + \color{blue}{\left(0 - C\right)}\right)}}{0 - B} \]

9. Taylor expanded in F around 0

   \[\leadsto \frac{\color{blue}{0}}{0 - B} \]
10. Step-by-step derivation

11. Simplified 3.9%

    \[\leadsto \frac{\color{blue}{0}}{0 - B} \]

12. Final simplification 3.9%

    \[\leadsto \frac{0}{B} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))