ABCF->ab-angle b

Percentage Accurate: 18.9% → 51.9%

Time: 20.5s

Alternatives: 12

Speedup: 10.4×

• could not determine a ground truth

  1. A = 6.703106555578432e-13
  2. B = -3.166633070444914e+58
  3. C = 1.2118580616112656e+129
  4. F = -6.944488085976757e-228

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := F \cdot \left(A + A\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{\left(t\_3 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, t\_1\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-1}{\frac{t\_3}{\sqrt{\left(A \cdot -8\right) \cdot t\_0} \cdot \sqrt{C}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{t\_3}{\sqrt{A \cdot C} \cdot \sqrt{-8 \cdot t\_0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B\_m} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B\_m} + \frac{-1}{F}}{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* F (+ A A)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma B_m B_m (* (* A C) -4.0))))
   (if (<= t_2 (- INFINITY))
     (/ -1.0 (* (sqrt (* -2.0 (/ C F))) (/ 1.0 (sqrt 2.0))))
     (if (<= t_2 -2e-161)
       (/
        (sqrt
         (*
          (* t_3 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) t_1))
       (if (<= t_2 0.0)
         (/ -1.0 (/ t_3 (* (sqrt (* (* A -8.0) t_0)) (sqrt C))))
         (if (<= t_2 INFINITY)
           (/ -1.0 (/ t_3 (* (sqrt (* A C)) (sqrt (* -8.0 t_0)))))
           (/
            -1.0
            (*
             (sqrt B_m)
             (sqrt
              (/ (+ (/ (* (+ A C) (/ -1.0 F)) B_m) (/ -1.0 F)) 2.0))))))))))

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) {
	double t_0 = F * (A + A);
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(B_m, B_m, ((A * C) * -4.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) * (1.0 / sqrt(2.0)));
	} else if (t_2 <= -2e-161) {
		tmp = sqrt(((t_3 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, t_1);
	} else if (t_2 <= 0.0) {
		tmp = -1.0 / (t_3 / (sqrt(((A * -8.0) * t_0)) * sqrt(C)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -1.0 / (t_3 / (sqrt((A * C)) * sqrt((-8.0 * t_0))));
	} else {
		tmp = -1.0 / (sqrt(B_m) * sqrt((((((A + C) * (-1.0 / F)) / B_m) + (-1.0 / F)) / 2.0)));
	}
	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(F * Float64(A + A))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(B_m, B_m, Float64(Float64(A * C) * -4.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) * Float64(1.0 / sqrt(2.0))));
	elseif (t_2 <= -2e-161)
		tmp = Float64(sqrt(Float64(Float64(t_3 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), t_1));
	elseif (t_2 <= 0.0)
		tmp = Float64(-1.0 / Float64(t_3 / Float64(sqrt(Float64(Float64(A * -8.0) * t_0)) * sqrt(C))));
	elseif (t_2 <= Inf)
		tmp = Float64(-1.0 / Float64(t_3 / Float64(sqrt(Float64(A * C)) * sqrt(Float64(-8.0 * t_0)))));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) * sqrt(Float64(Float64(Float64(Float64(Float64(A + C) * Float64(-1.0 / F)) / B_m) + Float64(-1.0 / F)) / 2.0))));
	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[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-161], N[(N[Sqrt[N[(N[(t$95$3 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(-1.0 / N[(t$95$3 / N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(-1.0 / N[(t$95$3 / N[(N[Sqrt[N[(A * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-8.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(A + C), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.2%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 3.2%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 29.0%

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

   8. Taylor expanded in A around -inf

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

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

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

      2. /-lowering-/.f64 34.5

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

   10. Simplified 34.5%

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

   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))) < -2.00000000000000006e-161

   1. Initial program 99.5%

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

      1. remove-double-neg N/A

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

      2. frac-2neg N/A

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

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

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

   4. Applied egg-rr 99.5%

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

   if -2.00000000000000006e-161 < (/.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 6.5%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 6.5%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 21.6

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

   7. Simplified 21.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. sqrt-prod N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. remove-double-neg N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. sqrt-lowering-sqrt.f64 29.8

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

   9. Applied egg-rr 29.8%

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

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

   1. Initial program 47.8%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 47.8%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 32.4

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

   7. Simplified 32.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

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

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

      7. pow1/2 N/A

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

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

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

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

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

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

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

      11. sub-neg N/A

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

      12. remove-double-neg N/A

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

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

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

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

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

      15. +-lowering-+.f64 36.7

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

   9. Applied egg-rr 36.7%

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

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

   1. Initial program 0.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 0.4%

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

   8. Taylor expanded in B around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 17.4

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

   10. Simplified 17.4%

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

       1. un-div-inv N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-/l* N/A

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

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

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

       6. pow1/2 N/A

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

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

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

       8. pow1/2 N/A

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

       9. sqrt-undiv N/A

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

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

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

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

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

   12. Applied egg-rr 18.1%

       \[\leadsto \frac{-1}{\color{blue}{\sqrt{B} \cdot \sqrt{\frac{\left(-\frac{\frac{1}{F} \cdot \left(A + C\right)}{B}\right) + \frac{-1}{F}}{2}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(A \cdot -8\right) \cdot \left(F \cdot \left(A + A\right)\right)} \cdot \sqrt{C}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{A \cdot C} \cdot \sqrt{-8 \cdot \left(F \cdot \left(A + A\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B} + \frac{-1}{F}}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(2 \cdot t\_0\right)\right) \cdot \mathsf{fma}\left(B\_m, \frac{B\_m}{C} \cdot -0.5, A + A\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B\_m} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B\_m} + \frac{-1}{F}}{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2000.0)
     (/
      (sqrt (* (* F (* 2.0 t_0)) (fma B_m (* (/ B_m C) -0.5) (+ A A))))
      (- t_0))
     (/
      -1.0
      (*
       (sqrt B_m)
       (sqrt (/ (+ (/ (* (+ A C) (/ -1.0 F)) B_m) (/ -1.0 F)) 2.0)))))))

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) {
	double t_0 = fma((A * C), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2000.0) {
		tmp = sqrt(((F * (2.0 * t_0)) * fma(B_m, ((B_m / C) * -0.5), (A + A)))) / -t_0;
	} else {
		tmp = -1.0 / (sqrt(B_m) * sqrt((((((A + C) * (-1.0 / F)) / B_m) + (-1.0 / F)) / 2.0)));
	}
	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(Float64(A * C), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2000.0)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(2.0 * t_0)) * fma(B_m, Float64(Float64(B_m / C) * -0.5), Float64(A + A)))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) * sqrt(Float64(Float64(Float64(Float64(Float64(A + C) * Float64(-1.0 / F)) / B_m) + Float64(-1.0 / F)) / 2.0))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2000.0], N[(N[Sqrt[N[(N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(B$95$m * N[(N[(B$95$m / C), $MachinePrecision] * -0.5), $MachinePrecision] + N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(A + C), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e3

   1. Initial program 24.9%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 18.9

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

   5. Simplified 18.9%

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

      1. frac-2neg N/A

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

      2. remove-double-neg N/A

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

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

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

   7. Applied egg-rr 18.9%

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

   if 2e3 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.6%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 15.6%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 22.3%

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

   8. Taylor expanded in B around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 24.3

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

   10. Simplified 24.3%

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

       1. un-div-inv N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-/l* N/A

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

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

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

       6. pow1/2 N/A

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

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

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

       8. pow1/2 N/A

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

       9. sqrt-undiv N/A

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

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

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

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

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

   12. Applied egg-rr 24.9%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)\right) \cdot \mathsf{fma}\left(B, \frac{B}{C} \cdot -0.5, A + A\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B} + \frac{-1}{F}}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.75 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 64000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;B\_m \leq 1.08 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{-1}{F \cdot \left(\sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)} - A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2}{\mathsf{fma}\left(B\_m, \frac{\left(A + C\right) \cdot \frac{-1}{F}}{B\_m}, \frac{B\_m}{-F}\right)}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.75e-170)
   (/
    (sqrt (* (* A C) (* -8.0 (* F (+ A A)))))
    (- (fma A (* C -4.0) (* B_m B_m))))
   (if (<= B_m 64000.0)
     (/ -1.0 (* (sqrt (* -2.0 (/ C F))) (/ 1.0 (sqrt 2.0))))
     (if (<= B_m 1.08e+126)
       (/
        -1.0
        (*
         (/ B_m (sqrt 2.0))
         (sqrt (/ -1.0 (* F (- (sqrt (fma A A (* B_m B_m))) A))))))
       (-
        (sqrt
         (/ 2.0 (fma B_m (/ (* (+ A C) (/ -1.0 F)) B_m) (/ B_m (- F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.75e-170) {
		tmp = sqrt(((A * C) * (-8.0 * (F * (A + A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (B_m <= 64000.0) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) * (1.0 / sqrt(2.0)));
	} else if (B_m <= 1.08e+126) {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt((-1.0 / (F * (sqrt(fma(A, A, (B_m * B_m))) - A)))));
	} else {
		tmp = -sqrt((2.0 / fma(B_m, (((A + C) * (-1.0 / F)) / B_m), (B_m / -F))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.75e-170)
		tmp = Float64(sqrt(Float64(Float64(A * C) * Float64(-8.0 * Float64(F * Float64(A + A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif (B_m <= 64000.0)
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) * Float64(1.0 / sqrt(2.0))));
	elseif (B_m <= 1.08e+126)
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(-1.0 / Float64(F * Float64(sqrt(fma(A, A, Float64(B_m * B_m))) - A))))));
	else
		tmp = Float64(-sqrt(Float64(2.0 / fma(B_m, Float64(Float64(Float64(A + C) * Float64(-1.0 / F)) / B_m), Float64(B_m / Float64(-F))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.75e-170], N[(N[Sqrt[N[(N[(A * C), $MachinePrecision] * N[(-8.0 * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 64000.0], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.08e+126], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(F * N[(N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(2.0 / N[(B$95$m * N[(N[(N[(A + C), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(B$95$m / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.74999999999999992e-170

   1. Initial program 19.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.9%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 12.0

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

   7. Simplified 12.0%

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

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

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

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

   9. Applied egg-rr 13.5%

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

   if 1.74999999999999992e-170 < B < 64000

   1. Initial program 30.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 29.9%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in A around -inf

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

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

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

      2. /-lowering-/.f64 19.7

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

   10. Simplified 19.7%

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

   if 64000 < B < 1.0799999999999999e126

   1. Initial program 44.1%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 44.1%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 47.9%

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

   8. Taylor expanded in C around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 46.0

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

   10. Simplified 46.0%

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

   if 1.0799999999999999e126 < B

   1. Initial program 0.5%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 0.5%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 6.5%

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

   8. Taylor expanded in B around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 64.5

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

   10. Simplified 64.5%

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

       1. div-inv N/A

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

       2. mul-1-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. clear-num N/A

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

       7. /-rgt-identity N/A

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

       8. sqrt-undiv N/A

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

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

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

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

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

       11. sub-neg N/A

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

   12. Applied egg-rr 65.2%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 64000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{-1}{F \cdot \left(\sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)} - A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2}{\mathsf{fma}\left(B, \frac{\left(A + C\right) \cdot \frac{-1}{F}}{B}, \frac{B}{-F}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.65 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 340000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B\_m} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B\_m} + \frac{-1}{F}}{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.65e-170)
   (/
    (sqrt (* (* A C) (* -8.0 (* F (+ A A)))))
    (- (fma A (* C -4.0) (* B_m B_m))))
   (if (<= B_m 340000.0)
     (/ -1.0 (* (sqrt (* -2.0 (/ C F))) (/ 1.0 (sqrt 2.0))))
     (/
      -1.0
      (*
       (sqrt B_m)
       (sqrt (/ (+ (/ (* (+ A C) (/ -1.0 F)) B_m) (/ -1.0 F)) 2.0)))))))

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) {
	double tmp;
	if (B_m <= 2.65e-170) {
		tmp = sqrt(((A * C) * (-8.0 * (F * (A + A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (B_m <= 340000.0) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) * (1.0 / sqrt(2.0)));
	} else {
		tmp = -1.0 / (sqrt(B_m) * sqrt((((((A + C) * (-1.0 / F)) / B_m) + (-1.0 / F)) / 2.0)));
	}
	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 <= 2.65e-170)
		tmp = Float64(sqrt(Float64(Float64(A * C) * Float64(-8.0 * Float64(F * Float64(A + A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif (B_m <= 340000.0)
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) * Float64(1.0 / sqrt(2.0))));
	else
		tmp = Float64(-1.0 / Float64(sqrt(B_m) * sqrt(Float64(Float64(Float64(Float64(Float64(A + C) * Float64(-1.0 / F)) / B_m) + Float64(-1.0 / F)) / 2.0))));
	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[B$95$m, 2.65e-170], N[(N[Sqrt[N[(N[(A * C), $MachinePrecision] * N[(-8.0 * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 340000.0], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[(A + C), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.65e-170

   1. Initial program 19.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.9%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 12.0

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

   7. Simplified 12.0%

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

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

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

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

   9. Applied egg-rr 13.5%

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

   if 2.65e-170 < B < 3.4e5

   1. Initial program 30.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 29.9%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in A around -inf

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

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

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

      2. /-lowering-/.f64 19.7

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

   10. Simplified 19.7%

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

   if 3.4e5 < B

   1. Initial program 18.7%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.7%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 23.8%

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

   8. Taylor expanded in B around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 53.8

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

   10. Simplified 53.8%

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

       1. un-div-inv N/A

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

       2. pow1/2 N/A

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

       3. unpow-prod-down N/A

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

       4. associate-/l* N/A

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

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

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

       6. pow1/2 N/A

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

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

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

       8. pow1/2 N/A

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

       9. sqrt-undiv N/A

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

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

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

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

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

   12. Applied egg-rr 57.5%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.65 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 340000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{B} \cdot \sqrt{\frac{\frac{\left(A + C\right) \cdot \frac{-1}{F}}{B} + \frac{-1}{F}}{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 24000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;B\_m \leq 8.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2}{\mathsf{fma}\left(B\_m, \frac{\left(A + C\right) \cdot \frac{-1}{F}}{B\_m}, \frac{B\_m}{-F}\right)}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.2e-170)
   (/
    (sqrt (* (* A C) (* -8.0 (* F (+ A A)))))
    (- (fma A (* C -4.0) (* B_m B_m))))
   (if (<= B_m 24000.0)
     (/ -1.0 (* (sqrt (* -2.0 (/ C F))) (/ 1.0 (sqrt 2.0))))
     (if (<= B_m 8.6e+124)
       (/ (sqrt (* 2.0 (* F (- A (sqrt (fma B_m B_m (* A A))))))) (- B_m))
       (-
        (sqrt
         (/ 2.0 (fma B_m (/ (* (+ A C) (/ -1.0 F)) B_m) (/ B_m (- F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.2e-170) {
		tmp = sqrt(((A * C) * (-8.0 * (F * (A + A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (B_m <= 24000.0) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) * (1.0 / sqrt(2.0)));
	} else if (B_m <= 8.6e+124) {
		tmp = sqrt((2.0 * (F * (A - sqrt(fma(B_m, B_m, (A * A))))))) / -B_m;
	} else {
		tmp = -sqrt((2.0 / fma(B_m, (((A + C) * (-1.0 / F)) / B_m), (B_m / -F))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.2e-170)
		tmp = Float64(sqrt(Float64(Float64(A * C) * Float64(-8.0 * Float64(F * Float64(A + A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif (B_m <= 24000.0)
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) * Float64(1.0 / sqrt(2.0))));
	elseif (B_m <= 8.6e+124)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A))))))) / Float64(-B_m));
	else
		tmp = Float64(-sqrt(Float64(2.0 / fma(B_m, Float64(Float64(Float64(A + C) * Float64(-1.0 / F)) / B_m), Float64(B_m / Float64(-F))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.2e-170], N[(N[Sqrt[N[(N[(A * C), $MachinePrecision] * N[(-8.0 * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 24000.0], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 8.6e+124], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[Sqrt[N[(2.0 / N[(B$95$m * N[(N[(N[(A + C), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] + N[(B$95$m / (-F)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.20000000000000015e-170

   1. Initial program 19.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.9%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 12.0

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

   7. Simplified 12.0%

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

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

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

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

   9. Applied egg-rr 13.5%

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

   if 2.20000000000000015e-170 < B < 24000

   1. Initial program 30.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 29.9%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in A around -inf

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

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

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

      2. /-lowering-/.f64 19.7

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

   10. Simplified 19.7%

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

   if 24000 < B < 8.6e124

   1. Initial program 44.1%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 44.1%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 45.8

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

   7. Simplified 45.8%

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

      1. associate-*l/ N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   9. Applied egg-rr 46.2%

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

   if 8.6e124 < B

   1. Initial program 0.5%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 0.5%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 6.5%

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

   8. Taylor expanded in B around inf

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

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

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

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

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

      3. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

      9. /-lowering-/.f64 64.5

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

   10. Simplified 64.5%

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

       1. div-inv N/A

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

       2. mul-1-neg N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. clear-num N/A

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

       7. /-rgt-identity N/A

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

       8. sqrt-undiv N/A

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

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

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

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

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

       11. sub-neg N/A

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

   12. Applied egg-rr 65.2%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 24000:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2}{\mathsf{fma}\left(B, \frac{\left(A + C\right) \cdot \frac{-1}{F}}{B}, \frac{B}{-F}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2}}\\ \mathbf{if}\;B\_m \leq 2.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 3500:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot t\_0}\\ \mathbf{elif}\;B\_m \leq 1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t\_0 \cdot \sqrt{\frac{B\_m}{-F}}}\\ \end{array} \end{array} \]

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 (/ 1.0 (sqrt 2.0))))
   (if (<= B_m 2.2e-170)
     (/
      (sqrt (* (* A C) (* -8.0 (* F (+ A A)))))
      (- (fma A (* C -4.0) (* B_m B_m))))
     (if (<= B_m 3500.0)
       (/ -1.0 (* (sqrt (* -2.0 (/ C F))) t_0))
       (if (<= B_m 1.95e+124)
         (/ (sqrt (* 2.0 (* F (- A (sqrt (fma B_m B_m (* A A))))))) (- B_m))
         (/ -1.0 (* t_0 (sqrt (/ B_m (- F))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 1.0 / sqrt(2.0);
	double tmp;
	if (B_m <= 2.2e-170) {
		tmp = sqrt(((A * C) * (-8.0 * (F * (A + A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (B_m <= 3500.0) {
		tmp = -1.0 / (sqrt((-2.0 * (C / F))) * t_0);
	} else if (B_m <= 1.95e+124) {
		tmp = sqrt((2.0 * (F * (A - sqrt(fma(B_m, B_m, (A * A))))))) / -B_m;
	} else {
		tmp = -1.0 / (t_0 * sqrt((B_m / -F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(1.0 / sqrt(2.0))
	tmp = 0.0
	if (B_m <= 2.2e-170)
		tmp = Float64(sqrt(Float64(Float64(A * C) * Float64(-8.0 * Float64(F * Float64(A + A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif (B_m <= 3500.0)
		tmp = Float64(-1.0 / Float64(sqrt(Float64(-2.0 * Float64(C / F))) * t_0));
	elseif (B_m <= 1.95e+124)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A))))))) / Float64(-B_m));
	else
		tmp = Float64(-1.0 / Float64(t_0 * sqrt(Float64(B_m / Float64(-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[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.2e-170], N[(N[Sqrt[N[(N[(A * C), $MachinePrecision] * N[(-8.0 * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3500.0], N[(-1.0 / N[(N[Sqrt[N[(-2.0 * N[(C / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.95e+124], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(-1.0 / N[(t$95$0 * N[Sqrt[N[(B$95$m / (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.20000000000000015e-170

   1. Initial program 19.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.9%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 12.0

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

   7. Simplified 12.0%

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

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

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

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

   9. Applied egg-rr 13.5%

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

   if 2.20000000000000015e-170 < B < 3500

   1. Initial program 30.0%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 29.9%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 17.2%

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

   8. Taylor expanded in A around -inf

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

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

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

      2. /-lowering-/.f64 19.7

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

   10. Simplified 19.7%

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

   if 3500 < B < 1.95e124

   1. Initial program 44.1%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 44.1%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 45.8

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

   7. Simplified 45.8%

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

      1. associate-*l/ N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   9. Applied egg-rr 46.2%

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

   if 1.95e124 < B

   1. Initial program 0.5%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 0.5%

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

   5. Taylor expanded in F around 0

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

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

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

   7. Simplified 6.5%

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

   8. Taylor expanded in B around inf

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

      1. associate-*r/ N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-lowering-neg.f64 63.3

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

   10. Simplified 63.3%

       \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3500:\\ \;\;\;\;\frac{-1}{\sqrt{-2 \cdot \frac{C}{F}} \cdot \frac{1}{\sqrt{2}}}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{B}{-F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 4.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{B\_m}{-F}}}\\ \end{array} \end{array} \]

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 (<= B_m 2.9e-81)
   (/
    (sqrt (* (* A C) (* -8.0 (* F (+ A A)))))
    (- (fma A (* C -4.0) (* B_m B_m))))
   (if (<= B_m 4.6e+130)
     (/ (sqrt (* 2.0 (* F (- A (sqrt (fma B_m B_m (* A A))))))) (- B_m))
     (/ -1.0 (* (/ 1.0 (sqrt 2.0)) (sqrt (/ B_m (- F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.9e-81) {
		tmp = sqrt(((A * C) * (-8.0 * (F * (A + A))))) / -fma(A, (C * -4.0), (B_m * B_m));
	} else if (B_m <= 4.6e+130) {
		tmp = sqrt((2.0 * (F * (A - sqrt(fma(B_m, B_m, (A * A))))))) / -B_m;
	} else {
		tmp = -1.0 / ((1.0 / sqrt(2.0)) * sqrt((B_m / -F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.9e-81)
		tmp = Float64(sqrt(Float64(Float64(A * C) * Float64(-8.0 * Float64(F * Float64(A + A))))) / Float64(-fma(A, Float64(C * -4.0), Float64(B_m * B_m))));
	elseif (B_m <= 4.6e+130)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A))))))) / Float64(-B_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(1.0 / sqrt(2.0)) * sqrt(Float64(B_m / Float64(-F)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.9e-81], N[(N[Sqrt[N[(N[(A * C), $MachinePrecision] * N[(-8.0 * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 4.6e+130], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(-1.0 / N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(B$95$m / (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.89999999999999989e-81

   1. Initial program 18.6%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 18.6%

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

   5. Taylor expanded in C around inf

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

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

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

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

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

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

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

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

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 12.3

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

   7. Simplified 12.3%

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

      1. associate-/r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*l/ N/A

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

      4. neg-mul-1 N/A

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

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

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

   9. Applied egg-rr 13.7%

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

   if 2.89999999999999989e-81 < B < 4.60000000000000042e130

   1. Initial program 44.3%

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

      1. distribute-frac-neg N/A

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

      2. neg-mul-1 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 37.9

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

   7. Simplified 37.9%

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

      1. associate-*l/ N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{\mathsf{neg}\left(B\right)}} \]

   9. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}} \]

   if 4.60000000000000042e130 < B

   1. Initial program 0.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      2. neg-mul-1 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      3. clear-num N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]

   7. Simplified 6.5%

      \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{-1}{\sqrt{\color{blue}{-1 \cdot \frac{B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-1 \cdot B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-1 \cdot B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-1}{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(B\right)}}{F}} \cdot \frac{1}{\sqrt{2}}} \]

      4. neg-lowering-neg.f64 63.3

         \[\leadsto \frac{-1}{\sqrt{\frac{\color{blue}{-B}}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   10. Simplified 63.3%

       \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot C\right) \cdot \left(-8 \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{B}{-F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{B\_m}{-F}}}\\ \end{array} \end{array} \]

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 (<= B_m 1.1e-130)
   (/ (sqrt (* -16.0 (* (* A A) (* C F)))) (- (fma (* A C) -4.0 (* B_m B_m))))
   (if (<= B_m 1.05e+130)
     (/ (sqrt (* 2.0 (* F (- A (sqrt (fma B_m B_m (* A A))))))) (- B_m))
     (/ -1.0 (* (/ 1.0 (sqrt 2.0)) (sqrt (/ B_m (- F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.1e-130) {
		tmp = sqrt((-16.0 * ((A * A) * (C * F)))) / -fma((A * C), -4.0, (B_m * B_m));
	} else if (B_m <= 1.05e+130) {
		tmp = sqrt((2.0 * (F * (A - sqrt(fma(B_m, B_m, (A * A))))))) / -B_m;
	} else {
		tmp = -1.0 / ((1.0 / sqrt(2.0)) * sqrt((B_m / -F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.1e-130)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F)))) / Float64(-fma(Float64(A * C), -4.0, Float64(B_m * B_m))));
	elseif (B_m <= 1.05e+130)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A))))))) / Float64(-B_m));
	else
		tmp = Float64(-1.0 / Float64(Float64(1.0 / sqrt(2.0)) * sqrt(Float64(B_m / Float64(-F)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.1e-130], N[(N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+130], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(-1.0 / N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(B$95$m / (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.0999999999999999e-130

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. *-lowering-*.f64 11.5

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 11.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)\right) \cdot \mathsf{fma}\left(B, \frac{B}{C} \cdot -0.5, A + A\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \color{blue}{\left(C \cdot F\right)}\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]

   10. Simplified 9.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]

   if 1.0999999999999999e-130 < B < 1.04999999999999995e130

   1. Initial program 42.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      2. neg-mul-1 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      3. clear-num N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. Applied egg-rr 42.3%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(A, A, {B}^{2}\right)}}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)}\right) \]

      13. *-lowering-*.f64 34.7

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)} \]

   7. Simplified 34.7%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{\mathsf{neg}\left(B\right)}} \]

   9. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}} \]

   if 1.04999999999999995e130 < B

   1. Initial program 0.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      2. neg-mul-1 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      3. clear-num N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]

   7. Simplified 6.5%

      \[\leadsto \frac{-1}{\color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}} \cdot \frac{1}{\sqrt{2}}}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{-1}{\sqrt{\color{blue}{-1 \cdot \frac{B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-1 \cdot B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-1 \cdot B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{-1}{\sqrt{\frac{\color{blue}{\mathsf{neg}\left(B\right)}}{F}} \cdot \frac{1}{\sqrt{2}}} \]

      4. neg-lowering-neg.f64 63.3

         \[\leadsto \frac{-1}{\sqrt{\frac{\color{blue}{-B}}{F}} \cdot \frac{1}{\sqrt{2}}} \]

   10. Simplified 63.3%

       \[\leadsto \frac{-1}{\sqrt{\color{blue}{\frac{-B}{F}}} \cdot \frac{1}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+130}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{1}{\sqrt{2}} \cdot \sqrt{\frac{B}{-F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.16 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\_m\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.16e-130)
   (/ (sqrt (* -16.0 (* (* A A) (* C F)))) (- (fma (* A C) -4.0 (* B_m B_m))))
   (* (sqrt (* F (- A B_m))) (- (/ (sqrt 2.0) B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.16e-130) {
		tmp = sqrt((-16.0 * ((A * A) * (C * F)))) / -fma((A * C), -4.0, (B_m * B_m));
	} else {
		tmp = sqrt((F * (A - B_m))) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.16e-130)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * A) * Float64(C * F)))) / Float64(-fma(Float64(A * C), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - B_m))) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.16e-130], N[(N[Sqrt[N[(-16.0 * N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.1600000000000001e-130

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around inf

      \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right) - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. *-lowering-*.f64 11.5

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Simplified 11.5%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}\right)\right)\right)}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{B \cdot B}{C} \cdot \frac{-1}{2} + A\right)\right)}}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   7. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)\right) \cdot \mathsf{fma}\left(B, \frac{B}{C} \cdot -0.5, A + A\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}} \]

   8. Taylor expanded in A around inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)}}{\mathsf{neg}\left(\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)} \]

      5. *-lowering-*.f64 9.6

         \[\leadsto \frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \color{blue}{\left(C \cdot F\right)}\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]

   10. Simplified 9.6%

       \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)} \]

   if 1.1600000000000001e-130 < B

   1. Initial program 25.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

      2. neg-mul-1 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

      3. clear-num N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. Applied egg-rr 25.1%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

   5. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(A, A, {B}^{2}\right)}}\right)}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)}\right) \]

      13. *-lowering-*.f64 24.1

          \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)} \]

   7. Simplified 24.1%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{B}\right)}\right) \]
   9. Step-by-step derivation

   10. Simplified 34.2%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{B}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.16 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{-\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{F \cdot \left(A - B\_m\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt (* F (- A B_m))) (- (/ (sqrt 2.0) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (A - B_m))) * -(sqrt(2.0) / B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * (a - b_m))) * -(sqrt(2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * (A - B_m))) * -(Math.sqrt(2.0) / B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * (A - B_m))) * -(math.sqrt(2.0) / B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(F * Float64(A - B_m))) * Float64(-Float64(sqrt(2.0) / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (A - B_m))) * -(sqrt(2.0) / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   3. clear-num N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

4. Applied egg-rr 20.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

5. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(A, A, {B}^{2}\right)}}\right)}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)}\right) \]

   13. *-lowering-*.f64 8.4

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)} \]

7. Simplified 8.4%

   \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}} \]

8. Taylor expanded in A around 0

   \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{B}\right)}\right) \]
9. Step-by-step derivation

10. Simplified 10.9%

    \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{B}\right)} \]

11. Final simplification 10.9%

    \[\leadsto \sqrt{F \cdot \left(A - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
12. Add Preprocessing

Alternative 11: 9.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{A \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((a * f)) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((A * F)) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((A * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * 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[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right)} \]

   2. neg-mul-1 N/A

      \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]

   3. clear-num N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{{B}^{2} - \left(4 \cdot A\right) \cdot C}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}} \]

4. Applied egg-rr 20.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}} \]

5. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\sqrt{2}}{B}} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\sqrt{2}}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \color{blue}{\sqrt{{A}^{2} + {B}^{2}}}\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{A \cdot A} + {B}^{2}}\right)}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\color{blue}{\mathsf{fma}\left(A, A, {B}^{2}\right)}}\right)}\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)}\right) \]

   13. *-lowering-*.f64 8.4

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, \color{blue}{B \cdot B}\right)}\right)} \]

7. Simplified 8.4%

   \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}} \]

8. Taylor expanded in A around -inf

   \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{A \cdot F}} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{A \cdot F}} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right)}}{B} \]

   6. rem-square-sqrt N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{\color{blue}{-1} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B} \]

   7. rem-square-sqrt N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{\color{blue}{-2}}{B} \]

   9. /-lowering-/.f64 2.1

      \[\leadsto \sqrt{A \cdot F} \cdot \color{blue}{\frac{-2}{B}} \]

10. Simplified 2.1%

    \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{-2}{B}} \]
11. Add Preprocessing

Alternative 12: 1.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* 2.0 (/ F B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 * (F / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 * (f / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 * (F / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((2.0 * (F / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 * Float64(F / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   2. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}}} \cdot \left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{2}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \sqrt{2}}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 1.8

       \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \color{blue}{\sqrt{2}}\right) \]

5. Simplified 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(--1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right)}\right)\right) \]

   2. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

   3. sqrt-unprod N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B} \cdot 2}} \]

   6. /-lowering-/.f64 1.8

      \[\leadsto \sqrt{\color{blue}{\frac{F}{B}} \cdot 2} \]

7. Applied egg-rr 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

8. Final simplification 1.8%

   \[\leadsto \sqrt{2 \cdot \frac{F}{B}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))