ABCF->ab-angle a

Percentage Accurate: 18.8% → 55.6%

Time: 53.3s

Alternatives: 23

Speedup: 5.8×

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: 18.8% 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: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ t_2 := t\_0 - B\_m \cdot B\_m\\ t_3 := \sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-127}:\\ \;\;\;\;\frac{t\_3 \cdot \left(\sqrt{2 \cdot \left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right)}{t\_2}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{t\_2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{t\_3 \cdot \sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_0 (pow B_m 2.0))))
        (t_2 (- t_0 (* B_m B_m)))
        (t_3 (sqrt (+ A (+ C (hypot B_m (- A C)))))))
   (if (<= t_1 -4e-127)
     (/
      (* t_3 (* (sqrt (* 2.0 (+ (* B_m B_m) (* A (* C -4.0))))) (sqrt F)))
      t_2)
     (if (<= t_1 0.0)
       (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) t_2)
       (if (<= t_1 INFINITY)
         (/ (* t_3 (sqrt (* 2.0 (* F (+ (* B_m B_m) (* -4.0 (* A C))))))) t_2)
         (* (sqrt F) (/ (sqrt (* 2.0 (+ A (hypot B_m A)))) (- 0.0 B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double t_2 = t_0 - (B_m * B_m);
	double t_3 = sqrt((A + (C + hypot(B_m, (A - C)))));
	double tmp;
	if (t_1 <= -4e-127) {
		tmp = (t_3 * (sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * sqrt(F))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_3 * sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / t_2;
	} else {
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_0 - Math.pow(B_m, 2.0));
	double t_2 = t_0 - (B_m * B_m);
	double t_3 = Math.sqrt((A + (C + Math.hypot(B_m, (A - C)))));
	double tmp;
	if (t_1 <= -4e-127) {
		tmp = (t_3 * (Math.sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * Math.sqrt(F))) / t_2;
	} else if (t_1 <= 0.0) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_3 * Math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / t_2;
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((2.0 * (A + Math.hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((A + C) + math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_0 - math.pow(B_m, 2.0))
	t_2 = t_0 - (B_m * B_m)
	t_3 = math.sqrt((A + (C + math.hypot(B_m, (A - C)))))
	tmp = 0
	if t_1 <= -4e-127:
		tmp = (t_3 * (math.sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * math.sqrt(F))) / t_2
	elif t_1 <= 0.0:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / t_2
	elif t_1 <= math.inf:
		tmp = (t_3 * math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / t_2
	else:
		tmp = math.sqrt(F) * (math.sqrt((2.0 * (A + math.hypot(B_m, A)))) / (0.0 - B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	t_2 = Float64(t_0 - Float64(B_m * B_m))
	t_3 = sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))
	tmp = 0.0
	if (t_1 <= -4e-127)
		tmp = Float64(Float64(t_3 * Float64(sqrt(Float64(2.0 * Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * sqrt(F))) / t_2);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / t_2);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_3 * sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))))) / t_2);
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(2.0 * Float64(A + hypot(B_m, A)))) / Float64(0.0 - B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((A + C) + sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_0 - (B_m ^ 2.0));
	t_2 = t_0 - (B_m * B_m);
	t_3 = sqrt((A + (C + hypot(B_m, (A - C)))));
	tmp = 0.0;
	if (t_1 <= -4e-127)
		tmp = (t_3 * (sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * sqrt(F))) / t_2;
	elseif (t_1 <= 0.0)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / t_2;
	elseif (t_1 <= Inf)
		tmp = (t_3 * sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / t_2;
	else
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $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$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -4e-127], N[(N[(t$95$3 * N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$3 * N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -4.0000000000000001e-127

   1. Initial program 44.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 50.1%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \left(\sqrt{\left(2 \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot F}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \left(\sqrt{2 \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{F}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \left(\sqrt{2 \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)} \cdot {F}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot C\right) \cdot -4\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(A \cdot \left(C \cdot -4\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(A \cdot \left(-4 \cdot C\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \left(-4 \cdot C\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \left(C \cdot -4\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      15. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right), \left(\sqrt{F}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      16. sqrt-lowering-sqrt.f64 72.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{+.f64}\left(C, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. Applied egg-rr 72.2%

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

   if -4.0000000000000001e-127 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 9.8%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 17.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 17.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 40.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 40.7%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   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 47.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 70.0%

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

   6. Applied egg-rr 86.8%

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

   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. Step-by-step derivation

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 1.4%

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

   6. Applied egg-rr 0.1%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 1.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 1.8%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{A \cdot A + B \cdot B}\right) \cdot F}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot {F}^{\frac{1}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       8. hypot-define N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       10. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right)\right) \]

       11. sqrt-lowering-sqrt.f64 27.4%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right)\right) \]

   11. Applied egg-rr 27.4%

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

       1. associate-*r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \color{blue}{\left(\mathsf{neg}\left(\sqrt{F}\right)\right)}\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}}{B}\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), B\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

       6. sqrt-unprod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

       10. hypot-define N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\left(\sqrt{F}\right)\right)\right) \]

       13. sqrt-lowering-sqrt.f64 27.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   13. Applied egg-rr 27.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e+58)
   (/
    (*
     (sqrt (+ A (+ C (hypot B_m (- A C)))))
     (sqrt (* 2.0 (* F (+ (* B_m B_m) (* -4.0 (* A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (sqrt F) (/ (sqrt (* 2.0 (+ A (hypot B_m A)))) (- 0.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e+58) {
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e+58) {
		tmp = (Math.sqrt((A + (C + Math.hypot(B_m, (A - C))))) * Math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((2.0 * (A + Math.hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.4e+58:
		tmp = (math.sqrt((A + (C + math.hypot(B_m, (A - C))))) * math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt(F) * (math.sqrt((2.0 * (A + math.hypot(B_m, A)))) / (0.0 - B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e+58)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(2.0 * Float64(A + hypot(B_m, A)))) / Float64(0.0 - B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.4e+58)
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e+58], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.4000000000000001e58

   1. Initial program 23.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 30.7%

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

   6. Applied egg-rr 35.3%

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

   if 4.4000000000000001e58 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.2%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.1%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{A \cdot A + B \cdot B}\right) \cdot F}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot {F}^{\frac{1}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       8. hypot-define N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       10. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right)\right) \]

       11. sqrt-lowering-sqrt.f64 74.1%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right)\right) \]

   11. Applied egg-rr 74.1%

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

       1. associate-*r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \color{blue}{\left(\mathsf{neg}\left(\sqrt{F}\right)\right)}\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}}{B}\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), B\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

       6. sqrt-unprod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

       10. hypot-define N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\left(\sqrt{F}\right)\right)\right) \]

       13. sqrt-lowering-sqrt.f64 74.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   13. Applied egg-rr 74.4%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.25 \cdot 10^{-165}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.25e-165)
   (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 7.5e+52)
     (/
      1.0
      (/
       (- (* 4.0 (* A C)) (* B_m B_m))
       (sqrt
        (*
         (+ (* B_m B_m) (* -4.0 (* A C)))
         (* (+ C (+ A (hypot B_m (- A C)))) (* 2.0 F))))))
     (* (sqrt F) (/ (sqrt (* 2.0 (+ A (hypot B_m A)))) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.25e-165) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.5e+52) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.25e-165) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 7.5e+52) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + Math.hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((2.0 * (A + Math.hypot(B_m, A)))) / (0.0 - B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.25e-165:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 7.5e+52:
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + math.hypot(B_m, (A - C)))) * (2.0 * F)))))
	else:
		tmp = math.sqrt(F) * (math.sqrt((2.0 * (A + math.hypot(B_m, A)))) / (0.0 - B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.25e-165)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 7.5e+52)
		tmp = Float64(1.0 / Float64(Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) * Float64(2.0 * F))))));
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(2.0 * Float64(A + hypot(B_m, A)))) / Float64(0.0 - B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.25e-165)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 7.5e+52)
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + hypot(B_m, (A - C)))) * (2.0 * F)))));
	else
		tmp = sqrt(F) * (sqrt((2.0 * (A + hypot(B_m, A)))) / (0.0 - B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.25e-165], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.5e+52], N[(1.0 / N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.24999999999999995e-165

   1. Initial program 20.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.4%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.24999999999999995e-165 < B < 7.49999999999999995e52

   1. Initial program 33.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 42.8%

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

   6. Applied egg-rr 43.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\left(A + \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} + C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\left(\left(A + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) + C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(A + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A - C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, \left(A - C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 43.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 43.8%

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

   if 7.49999999999999995e52 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.7%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 14.6%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 14.6%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{A \cdot A + B \cdot B}\right) \cdot F}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{\left(A + \sqrt{B \cdot B + A \cdot A}\right) \cdot F}\right)\right)\right) \]

       3. sqrt-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot \sqrt{F}\right)\right)\right) \]

       4. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}} \cdot {F}^{\frac{1}{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\left(\sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A + \sqrt{B \cdot B + A \cdot A}\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       8. hypot-define N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left({F}^{\frac{1}{2}}\right)\right)\right)\right) \]

       10. pow1/2 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \left(\sqrt{F}\right)\right)\right)\right) \]

       11. sqrt-lowering-sqrt.f64 70.3%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right), \mathsf{sqrt.f64}\left(F\right)\right)\right)\right) \]

   11. Applied egg-rr 70.3%

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

       1. associate-*r* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), \color{blue}{\left(\mathsf{neg}\left(\sqrt{F}\right)\right)}\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}}{B}\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{A + \sqrt{B \cdot B + A \cdot A}}\right), B\right), \left(\mathsf{neg}\left(\color{blue}{\sqrt{F}}\right)\right)\right) \]

       6. sqrt-unprod N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{\color{blue}{F}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + A \cdot A}\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

       10. hypot-define N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A\right)\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \left(\mathsf{neg}\left(\sqrt{F}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\left(\sqrt{F}\right)\right)\right) \]

       13. sqrt-lowering-sqrt.f64 70.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, A\right)\right)\right)\right), B\right), \mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(F\right)\right)\right) \]

   13. Applied egg-rr 70.5%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-165}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \frac{\sqrt{2 \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.1e-165)
   (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 2.45e+67)
     (/
      1.0
      (/
       (- (* 4.0 (* A C)) (* B_m B_m))
       (sqrt
        (*
         (+ (* B_m B_m) (* -4.0 (* A C)))
         (* (+ C (+ A (hypot B_m (- A C)))) (* 2.0 F))))))
     (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.1e-165) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 2.45e+67) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.1e-165) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 2.45e+67) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + Math.hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.1e-165:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 2.45e+67:
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + math.hypot(B_m, (A - C)))) * (2.0 * F)))))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.1e-165)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 2.45e+67)
		tmp = Float64(1.0 / Float64(Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) * Float64(2.0 * F))))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.1e-165)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 2.45e+67)
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((C + (A + hypot(B_m, (A - C)))) * (2.0 * F)))));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.1e-165], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.45e+67], N[(1.0 / N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.0999999999999999e-165

   1. Initial program 20.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.4%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.0999999999999999e-165 < B < 2.44999999999999995e67

   1. Initial program 31.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 40.2%

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

   6. Applied egg-rr 40.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\left(A + \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} + C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\left(\left(A + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right) + C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(A + \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \left(\mathsf{hypot}\left(B, A - C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, \left(A - C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 41.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right), C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 41.2%

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

   if 2.44999999999999995e67 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.8e-160)
   (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 3.1e+66)
     (/
      1.0
      (/
       (- (* 4.0 (* A C)) (* B_m B_m))
       (sqrt
        (*
         (+ (* B_m B_m) (* -4.0 (* A C)))
         (* (+ A (+ C (hypot B_m (- A C)))) (* 2.0 F))))))
     (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.8e-160) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 3.1e+66) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.8e-160) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 3.1e+66) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + Math.hypot(B_m, (A - C)))) * (2.0 * F)))));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.8e-160:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 3.1e+66:
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + math.hypot(B_m, (A - C)))) * (2.0 * F)))))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.8e-160)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 3.1e+66)
		tmp = Float64(1.0 / Float64(Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(2.0 * F))))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.8e-160)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 3.1e+66)
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + hypot(B_m, (A - C)))) * (2.0 * F)))));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.8e-160], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.1e+66], N[(1.0 / N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.79999999999999979e-160

   1. Initial program 20.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.4%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.79999999999999979e-160 < B < 3.10000000000000019e66

   1. Initial program 31.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 40.2%

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

   6. Applied egg-rr 40.9%

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

   if 3.10000000000000019e66 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

Alternative 6: 46.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.02 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.2e-148)
   (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 1.02e+67)
     (*
      (sqrt
       (*
        (+ (* B_m B_m) (* -4.0 (* A C)))
        (* (+ A (+ C (hypot B_m (- A C)))) (* 2.0 F))))
      (/ 1.0 (- (* 4.0 (* A C)) (* B_m B_m))))
     (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-148) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.02e+67) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + hypot(B_m, (A - C)))) * (2.0 * F)))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-148) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 1.02e+67) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + Math.hypot(B_m, (A - C)))) * (2.0 * F)))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.2e-148:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 1.02e+67:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + math.hypot(B_m, (A - C)))) * (2.0 * F)))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.2e-148)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 1.02e+67)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(2.0 * F)))) * Float64(1.0 / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.2e-148)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 1.02e+67)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((A + (C + hypot(B_m, (A - C)))) * (2.0 * F)))) * (1.0 / ((4.0 * (A * C)) - (B_m * B_m)));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.2e-148], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.02e+67], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.1999999999999999e-148

   1. Initial program 20.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.2%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.2%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.1999999999999999e-148 < B < 1.02000000000000002e67

   1. Initial program 32.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 41.6%

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

   6. Applied egg-rr 42.2%

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

   if 1.02000000000000002e67 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{t\_0}\\ \mathbf{elif}\;B\_m \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 1.9e-160)
     (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) t_0)
     (if (<= B_m 3.2e+66)
       (/
        (sqrt
         (*
          (+ (* B_m B_m) (* A (* C -4.0)))
          (* (* 2.0 F) (+ (+ A C) (hypot B_m (- A C))))))
        t_0)
       (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 1.9e-160) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / t_0;
	} else if (B_m <= 3.2e+66) {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 1.9e-160) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / t_0;
	} else if (B_m <= 3.2e+66) {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * ((2.0 * F) * ((A + C) + Math.hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 1.9e-160:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / t_0
	elif B_m <= 3.2e+66:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * ((2.0 * F) * ((A + C) + math.hypot(B_m, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.9e-160)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / t_0);
	elseif (B_m <= 3.2e+66)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(Float64(2.0 * F) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 1.9e-160)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / t_0;
	elseif (B_m <= 3.2e+66)
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / t_0;
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.9e-160], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 3.2e+66], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.8999999999999999e-160

   1. Initial program 20.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.4%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.8999999999999999e-160 < B < 3.2e66

   1. Initial program 31.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 40.2%

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

   if 3.2e66 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.9 \cdot 10^{-160}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 8.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{t\_0}\\ \mathbf{elif}\;B\_m \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 3.1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 8.2e-144)
     (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) t_0)
     (if (<= B_m 6.2e-70)
       (/
        1.0
        (/
         (- (* 4.0 (* A C)) (* B_m B_m))
         (sqrt (* (* 4.0 C) (* F (+ (* B_m B_m) (* A (* C -4.0))))))))
       (if (<= B_m 3.1e+66)
         (/
          (sqrt (* (* B_m B_m) (* (* 2.0 F) (+ (+ A C) (hypot B_m (- A C))))))
          t_0)
         (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 8.2e-144) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / t_0;
	} else if (B_m <= 6.2e-70) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	} else if (B_m <= 3.1e+66) {
		tmp = sqrt(((B_m * B_m) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 8.2e-144) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / t_0;
	} else if (B_m <= 6.2e-70) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / Math.sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	} else if (B_m <= 3.1e+66) {
		tmp = Math.sqrt(((B_m * B_m) * ((2.0 * F) * ((A + C) + Math.hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 8.2e-144:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / t_0
	elif B_m <= 6.2e-70:
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / math.sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))))
	elif B_m <= 3.1e+66:
		tmp = math.sqrt(((B_m * B_m) * ((2.0 * F) * ((A + C) + math.hypot(B_m, (A - C)))))) / t_0
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 8.2e-144)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / t_0);
	elseif (B_m <= 6.2e-70)
		tmp = Float64(1.0 / Float64(Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)) / sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))))));
	elseif (B_m <= 3.1e+66)
		tmp = Float64(sqrt(Float64(Float64(B_m * B_m) * Float64(Float64(2.0 * F) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C)))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 8.2e-144)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / t_0;
	elseif (B_m <= 6.2e-70)
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	elseif (B_m <= 3.1e+66)
		tmp = sqrt(((B_m * B_m) * ((2.0 * F) * ((A + C) + hypot(B_m, (A - C)))))) / t_0;
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.2e-144], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 6.2e-70], N[(1.0 / N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.1e+66], N[(N[Sqrt[N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 8.2e-144

   1. Initial program 20.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.2%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.2%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 8.2e-144 < B < 6.2e-70

   1. Initial program 20.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 33.9%

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

   6. Applied egg-rr 34.8%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 29.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 29.9%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot C\right) \cdot 2\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(C \cdot 2\right) \cdot 2\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(C \cdot \left(2 \cdot 2\right)\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(C \cdot 4\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot C\right) \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(A \cdot \left(C \cdot -4\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \left(C \cdot -4\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 34.3%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 34.3%

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

   if 6.2e-70 < B < 3.10000000000000019e66

   1. Initial program 40.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 46.2%

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

   5. Taylor expanded in B around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left({B}^{2}\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 36.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(A, C\right), \mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 36.2%

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

   if 3.10000000000000019e66 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{-144}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{t\_0}\\ \mathbf{elif}\;B\_m \leq 12500000000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 6.5e-157)
     (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) t_0)
     (if (<= B_m 12500000000.0)
       (/
        (*
         (sqrt (* 2.0 (* F (+ (* B_m B_m) (* -4.0 (* A C))))))
         (sqrt (* 2.0 C)))
        t_0)
       (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6.5e-157) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / t_0;
	} else if (B_m <= 12500000000.0) {
		tmp = (sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (b_m <= 6.5d-157) then
        tmp = (((c * (f * (-16.0d0))) ** 0.5d0) * abs(a)) / t_0
    else if (b_m <= 12500000000.0d0) then
        tmp = (sqrt((2.0d0 * (f * ((b_m * b_m) + ((-4.0d0) * (a * c)))))) * sqrt((2.0d0 * c))) / t_0
    else
        tmp = (sqrt((2.0d0 * f)) / (-1.0d0)) * (sqrt(b_m) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6.5e-157) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / t_0;
	} else if (B_m <= 12500000000.0) {
		tmp = (Math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C)))))) * Math.sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 6.5e-157:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / t_0
	elif B_m <= 12500000000.0:
		tmp = (math.sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C)))))) * math.sqrt((2.0 * C))) / t_0
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6.5e-157)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / t_0);
	elseif (B_m <= 12500000000.0)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) * sqrt(Float64(2.0 * C))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 6.5e-157)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / t_0;
	elseif (B_m <= 12500000000.0)
		tmp = (sqrt((2.0 * (F * ((B_m * B_m) + (-4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_0;
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.5e-157], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 12500000000.0], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.5000000000000002e-157

   1. Initial program 20.4%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.4%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 6.5000000000000002e-157 < B < 1.25e10

   1. Initial program 31.8%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 41.2%

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

   6. Applied egg-rr 43.5%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot C\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 31.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, C\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 31.7%

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

   if 1.25e10 < B

   1. Initial program 9.9%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 11.6%

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

   6. Applied egg-rr 20.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 19.6%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 19.6%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 46.4%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 62.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 62.2%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 12500000000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.25 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(F \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.25e+66)
   (/
    (sqrt
     (*
      2.0
      (*
       (+ A (+ C (hypot B_m (- A C))))
       (* F (+ (* B_m B_m) (* -4.0 (* A C)))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.25e+66) {
		tmp = sqrt((2.0 * ((A + (C + hypot(B_m, (A - C)))) * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.25e+66) {
		tmp = Math.sqrt((2.0 * ((A + (C + Math.hypot(B_m, (A - C)))) * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.25e+66:
		tmp = math.sqrt((2.0 * ((A + (C + math.hypot(B_m, (A - C)))) * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.25e+66)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(F * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.25e+66)
		tmp = sqrt((2.0 * ((A + (C + hypot(B_m, (A - C)))) * (F * ((B_m * B_m) + (-4.0 * (A * C))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.25e+66], N[(N[Sqrt[N[(2.0 * N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.2500000000000001e66

   1. Initial program 23.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 30.5%

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

   6. Applied egg-rr 30.4%

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

   if 3.2500000000000001e66 < B

   1. Initial program 2.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.6%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 15.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 15.3%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 51.0%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 72.2%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.25 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 0.0059:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.6e-147)
   (/ (* (pow (* C (* F -16.0)) 0.5) (fabs A)) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= B_m 0.0059)
     (/
      1.0
      (/
       (- (* 4.0 (* A C)) (* B_m B_m))
       (sqrt (* (* 4.0 C) (* F (+ (* B_m B_m) (* A (* C -4.0))))))))
     (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.6e-147) {
		tmp = (pow((C * (F * -16.0)), 0.5) * fabs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 0.0059) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 7.6d-147) then
        tmp = (((c * (f * (-16.0d0))) ** 0.5d0) * abs(a)) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (b_m <= 0.0059d0) then
        tmp = 1.0d0 / (((4.0d0 * (a * c)) - (b_m * b_m)) / sqrt(((4.0d0 * c) * (f * ((b_m * b_m) + (a * (c * (-4.0d0))))))))
    else
        tmp = (sqrt((2.0d0 * f)) / (-1.0d0)) * (sqrt(b_m) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.6e-147) {
		tmp = (Math.pow((C * (F * -16.0)), 0.5) * Math.abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (B_m <= 0.0059) {
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / Math.sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.6e-147:
		tmp = (math.pow((C * (F * -16.0)), 0.5) * math.fabs(A)) / (((4.0 * A) * C) - (B_m * B_m))
	elif B_m <= 0.0059:
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / math.sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.6e-147)
		tmp = Float64(Float64((Float64(C * Float64(F * -16.0)) ^ 0.5) * abs(A)) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (B_m <= 0.0059)
		tmp = Float64(1.0 / Float64(Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)) / sqrt(Float64(Float64(4.0 * C) * Float64(F * Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.6e-147)
		tmp = (((C * (F * -16.0)) ^ 0.5) * abs(A)) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (B_m <= 0.0059)
		tmp = 1.0 / (((4.0 * (A * C)) - (B_m * B_m)) / sqrt(((4.0 * C) * (F * ((B_m * B_m) + (A * (C * -4.0)))))));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.6e-147], N[(N[(N[Power[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Abs[A], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 0.0059], N[(1.0 / N[(N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.60000000000000055e-147

   1. Initial program 20.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 26.2%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 10.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(-16 \cdot \left(C \cdot F\right)\right) \cdot \left(A \cdot A\right)\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}} \cdot {\left(A \cdot A\right)}^{\frac{1}{2}}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(C \cdot F\right) \cdot -16\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(C \cdot \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \left(F \cdot -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left({\left(A \cdot A\right)}^{\frac{1}{2}}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\sqrt{A \cdot A}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      13. rem-sqrt-square N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \left(\left|A\right|\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      14. fabs-lowering-fabs.f64 17.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, -16\right)\right), \frac{1}{2}\right), \mathsf{fabs.f64}\left(A\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), \color{blue}{C}\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.2%

      \[\leadsto \frac{\color{blue}{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.60000000000000055e-147 < B < 0.00589999999999999986

   1. Initial program 34.8%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 45.1%

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

   6. Applied egg-rr 45.6%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 30.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 30.4%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot F\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(2 \cdot C\right) \cdot 2\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(C \cdot 2\right) \cdot 2\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(C \cdot \left(2 \cdot 2\right)\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(C \cdot 4\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(B \cdot B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right)\right)\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot C\right) \cdot -4\right)\right)\right)\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(A \cdot \left(C \cdot -4\right)\right)\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \left(C \cdot -4\right)\right)\right)\right)\right)\right)\right)\right) \]

       17. *-lowering-*.f64 30.5%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, 4\right), \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 30.5%

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

   if 0.00589999999999999986 < B

   1. Initial program 9.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 11.4%

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

   6. Applied egg-rr 22.7%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 19.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 19.1%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 45.2%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 60.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 60.6%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{-147}:\\ \;\;\;\;\frac{{\left(C \cdot \left(F \cdot -16\right)\right)}^{0.5} \cdot \left|A\right|}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 0.0059:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F}{B\_m}} \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.8e-307)
   (/
    (sqrt (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))))
    (- (* A (* 4.0 C)) (* B_m B_m)))
   (if (<= F 2.9e+28)
     (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))
     (- 0.0 (* (sqrt (/ F B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.8e-307) {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	} else if (F <= 2.9e+28) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	} else {
		tmp = 0.0 - (sqrt((F / B_m)) * sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 2.8d-307) then
        tmp = sqrt((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c)))) / ((a * (4.0d0 * c)) - (b_m * b_m))
    else if (f <= 2.9d+28) then
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    else
        tmp = 0.0d0 - (sqrt((f / b_m)) * sqrt(2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.8e-307) {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	} else if (F <= 2.9e+28) {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	} else {
		tmp = 0.0 - (Math.sqrt((F / B_m)) * Math.sqrt(2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.8e-307:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m))
	elif F <= 2.9e+28:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	else:
		tmp = 0.0 - (math.sqrt((F / B_m)) * math.sqrt(2.0))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.8e-307)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C)))) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)));
	elseif (F <= 2.9e+28)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	else
		tmp = Float64(0.0 - Float64(sqrt(Float64(F / B_m)) * sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.8e-307)
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	elseif (F <= 2.9e+28)
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	else
		tmp = 0.0 - (sqrt((F / B_m)) * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.8e-307], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e+28], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.8e-307

   1. Initial program 34.6%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 58.4%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 37.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 37.2%

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

       1. clear-num N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)}\right) \]

   11. Applied egg-rr 37.3%

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

   if 2.8e-307 < F < 2.9000000000000001e28

   1. Initial program 16.0%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 19.9%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 13.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 13.2%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 25.3%

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

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 25.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 25.5%

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

   if 2.9000000000000001e28 < F

   1. Initial program 17.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left(\sqrt{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left(\sqrt{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\sqrt{2}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 18.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   5. Simplified 18.8%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \sqrt{\frac{F}{B}} \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.00145:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B\_m}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.00145)
   (/
    (sqrt (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))))
    (- (* A (* 4.0 C)) (* B_m B_m)))
   (* (/ (sqrt (* 2.0 F)) -1.0) (/ (sqrt B_m) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.00145) {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 0.00145d0) then
        tmp = sqrt((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c)))) / ((a * (4.0d0 * c)) - (b_m * b_m))
    else
        tmp = (sqrt((2.0d0 * f)) / (-1.0d0)) * (sqrt(b_m) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.00145) {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	} else {
		tmp = (Math.sqrt((2.0 * F)) / -1.0) * (Math.sqrt(B_m) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 0.00145:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m))
	else:
		tmp = (math.sqrt((2.0 * F)) / -1.0) * (math.sqrt(B_m) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.00145)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C)))) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * F)) / -1.0) * Float64(sqrt(B_m) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 0.00145)
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	else
		tmp = (sqrt((2.0 * F)) / -1.0) * (sqrt(B_m) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.00145], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[B$95$m], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00145

   1. Initial program 23.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 30.2%

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

   6. Applied egg-rr 30.9%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 19.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 19.7%

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

       1. clear-num N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)}\right) \]

   11. Applied egg-rr 19.7%

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

   if 0.00145 < B

   1. Initial program 9.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 11.4%

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

   6. Applied egg-rr 22.7%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 19.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 19.1%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 45.2%

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

       1. associate-*l/ N/A

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

       2. distribute-neg-frac2 N/A

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

       3. sqrt-prod N/A

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

       4. pow1/2 N/A

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

       5. associate-*r* N/A

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

       6. neg-mul-1 N/A

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

       7. times-frac N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F}}{-1}\right), \color{blue}{\left(\frac{{B}^{\frac{1}{2}}}{B}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F}\right), -1\right), \left(\frac{\color{blue}{{B}^{\frac{1}{2}}}}{B}\right)\right) \]

       10. sqrt-unprod N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2 \cdot F}\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot F\right)\right), -1\right), \left(\frac{{\color{blue}{B}}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \left(\frac{{B}^{\frac{1}{2}}}{B}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left({B}^{\frac{1}{2}}\right), \color{blue}{B}\right)\right) \]

       14. pow1/2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\left(\sqrt{B}\right), B\right)\right) \]

       15. sqrt-lowering-sqrt.f64 60.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, F\right)\right), -1\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(B\right), B\right)\right) \]

   14. Applied egg-rr 60.6%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00145:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-1} \cdot \frac{\sqrt{B}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m\\ t_1 := \frac{B\_m \cdot B\_m}{A}\\ \mathbf{if}\;A \leq -7.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C + -0.5 \cdot t\_1\right)\right)}}}\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{A}{t\_0} \cdot \sqrt{F \cdot \left(C \cdot -16 + 2 \cdot \left(-2 \cdot \frac{C \cdot \left(B\_m \cdot B\_m\right)}{A \cdot A} + 2 \cdot t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* 4.0 (* A C)) (* B_m B_m))) (t_1 (/ (* B_m B_m) A)))
   (if (<= A -7.5e+53)
     (/
      1.0
      (/
       t_0
       (sqrt
        (*
         (+ (* B_m B_m) (* -4.0 (* A C)))
         (* (* 2.0 F) (+ (* 2.0 C) (* -0.5 t_1)))))))
     (if (<= A 3.4e-15)
       (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))
       (*
        (/ A t_0)
        (sqrt
         (*
          F
          (+
           (* C -16.0)
           (*
            2.0
            (+ (* -2.0 (/ (* C (* B_m B_m)) (* A A))) (* 2.0 t_1)))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double t_1 = (B_m * B_m) / A;
	double tmp;
	if (A <= -7.5e+53) {
		tmp = 1.0 / (t_0 / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * C) + (-0.5 * t_1))))));
	} else if (A <= 3.4e-15) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	} else {
		tmp = (A / t_0) * sqrt((F * ((C * -16.0) + (2.0 * ((-2.0 * ((C * (B_m * B_m)) / (A * A))) + (2.0 * t_1))))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * (a * c)) - (b_m * b_m)
    t_1 = (b_m * b_m) / a
    if (a <= (-7.5d+53)) then
        tmp = 1.0d0 / (t_0 / sqrt((((b_m * b_m) + ((-4.0d0) * (a * c))) * ((2.0d0 * f) * ((2.0d0 * c) + ((-0.5d0) * t_1))))))
    else if (a <= 3.4d-15) then
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    else
        tmp = (a / t_0) * sqrt((f * ((c * (-16.0d0)) + (2.0d0 * (((-2.0d0) * ((c * (b_m * b_m)) / (a * a))) + (2.0d0 * t_1))))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double t_1 = (B_m * B_m) / A;
	double tmp;
	if (A <= -7.5e+53) {
		tmp = 1.0 / (t_0 / Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * C) + (-0.5 * t_1))))));
	} else if (A <= 3.4e-15) {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	} else {
		tmp = (A / t_0) * Math.sqrt((F * ((C * -16.0) + (2.0 * ((-2.0 * ((C * (B_m * B_m)) / (A * A))) + (2.0 * t_1))))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * (A * C)) - (B_m * B_m)
	t_1 = (B_m * B_m) / A
	tmp = 0
	if A <= -7.5e+53:
		tmp = 1.0 / (t_0 / math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * C) + (-0.5 * t_1))))))
	elif A <= 3.4e-15:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	else:
		tmp = (A / t_0) * math.sqrt((F * ((C * -16.0) + (2.0 * ((-2.0 * ((C * (B_m * B_m)) / (A * A))) + (2.0 * t_1))))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))
	t_1 = Float64(Float64(B_m * B_m) / A)
	tmp = 0.0
	if (A <= -7.5e+53)
		tmp = Float64(1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(Float64(2.0 * C) + Float64(-0.5 * t_1)))))));
	elseif (A <= 3.4e-15)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	else
		tmp = Float64(Float64(A / t_0) * sqrt(Float64(F * Float64(Float64(C * -16.0) + Float64(2.0 * Float64(Float64(-2.0 * Float64(Float64(C * Float64(B_m * B_m)) / Float64(A * A))) + Float64(2.0 * t_1)))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * (A * C)) - (B_m * B_m);
	t_1 = (B_m * B_m) / A;
	tmp = 0.0;
	if (A <= -7.5e+53)
		tmp = 1.0 / (t_0 / sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * ((2.0 * C) + (-0.5 * t_1))))));
	elseif (A <= 3.4e-15)
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	else
		tmp = (A / t_0) * sqrt((F * ((C * -16.0) + (2.0 * ((-2.0 * ((C * (B_m * B_m)) / (A * A))) + (2.0 * t_1))))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, If[LessEqual[A, -7.5e+53], N[(1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(2.0 * C), $MachinePrecision] + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 3.4e-15], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(A / t$95$0), $MachinePrecision] * N[Sqrt[N[(F * N[(N[(C * -16.0), $MachinePrecision] + N[(2.0 * N[(N[(-2.0 * N[(N[(C * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -7.4999999999999997e53

   1. Initial program 1.2%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.3%

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

   6. Applied egg-rr 2.3%

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

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right), \left(2 \cdot C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{{B}^{2}}{A}\right)\right), \left(2 \cdot C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left({B}^{2}\right), A\right)\right), \left(2 \cdot C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(B \cdot B\right), A\right)\right), \left(2 \cdot C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(B, B\right), A\right)\right), \left(2 \cdot C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 29.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(B, B\right), A\right)\right), \mathsf{*.f64}\left(2, C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 29.1%

      \[\leadsto \frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(-0.5 \cdot \frac{B \cdot B}{A} + 2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}} \]

   if -7.4999999999999997e53 < A < 3.4e-15

   1. Initial program 24.5%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 30.2%

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

   6. Applied egg-rr 39.9%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 14.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 14.7%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 24.6%

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

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 24.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 24.8%

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

   if 3.4e-15 < A

   1. Initial program 25.7%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 35.5%

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

   5. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 19.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)}{A} + \frac{F \cdot \left(-2 \cdot \left(\left(B \cdot B\right) \cdot C\right) + \left(B \cdot B\right) \cdot \left(0 \cdot C\right)\right)}{A \cdot A}\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{A}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \cdot \sqrt{F \cdot \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{A}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\right), \color{blue}{\left(\sqrt{F \cdot \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \left(4 \cdot \left(A \cdot C\right) - {B}^{2}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\left(4 \cdot \left(A \cdot C\right)\right), \left({B}^{2}\right)\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \left(A \cdot C\right)\right), \left({B}^{2}\right)\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{-16 \cdot C} + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \left({B}^{2}\right)\right)\right), \left(\sqrt{F \cdot \left(-16 \cdot \color{blue}{C} + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{F \cdot \left(-16 \cdot C + \color{blue}{2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \left(\sqrt{F \cdot \left(-16 \cdot C + \color{blue}{2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\left(-16 \cdot C\right), \left(2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, C\right), \left(2 \cdot \left(-2 \cdot \frac{{B}^{2} \cdot C}{{A}^{2}} + 2 \cdot \frac{{B}^{2}}{A}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 31.4%

       \[\leadsto \color{blue}{\frac{A}{4 \cdot \left(A \cdot C\right) - B \cdot B} \cdot \sqrt{F \cdot \left(-16 \cdot C + 2 \cdot \left(-2 \cdot \frac{\left(B \cdot B\right) \cdot C}{A \cdot A} + 2 \cdot \frac{B \cdot B}{A}\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -7.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}}\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{A}{4 \cdot \left(A \cdot C\right) - B \cdot B} \cdot \sqrt{F \cdot \left(C \cdot -16 + 2 \cdot \left(-2 \cdot \frac{C \cdot \left(B \cdot B\right)}{A \cdot A} + 2 \cdot \frac{B \cdot B}{A}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -6.5 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}}{{\left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -6.5e+197)
   (/ (sqrt (* (* F -16.0) (* A (* A C)))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= C 1.3e-72)
     (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))
     (/
      (/ 1.0 (- (* A (* 4.0 C)) (* B_m B_m)))
      (pow (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -6.5e+197) {
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 1.3e-72) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	} else {
		tmp = (1.0 / ((A * (4.0 * C)) - (B_m * B_m))) / pow((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C))), -0.5);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-6.5d+197)) then
        tmp = sqrt(((f * (-16.0d0)) * (a * (a * c)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (c <= 1.3d-72) then
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    else
        tmp = (1.0d0 / ((a * (4.0d0 * c)) - (b_m * b_m))) / ((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c))) ** (-0.5d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -6.5e+197) {
		tmp = Math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 1.3e-72) {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	} else {
		tmp = (1.0 / ((A * (4.0 * C)) - (B_m * B_m))) / Math.pow((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C))), -0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -6.5e+197:
		tmp = math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif C <= 1.3e-72:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	else:
		tmp = (1.0 / ((A * (4.0 * C)) - (B_m * B_m))) / math.pow((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C))), -0.5)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -6.5e+197)
		tmp = Float64(sqrt(Float64(Float64(F * -16.0) * Float64(A * Float64(A * C)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (C <= 1.3e-72)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m))) / (Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C))) ^ -0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -6.5e+197)
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (C <= 1.3e-72)
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	else
		tmp = (1.0 / ((A * (4.0 * C)) - (B_m * B_m))) / ((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C))) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -6.5e+197], N[(N[Sqrt[N[(N[(F * -16.0), $MachinePrecision] * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.3e-72], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -6.49999999999999952e197

   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. Step-by-step derivation

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 2.1%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot A\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 27.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 27.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -6.49999999999999952e197 < C < 1.29999999999999998e-72

   1. Initial program 22.1%

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

      1. distribute-frac-neg N/A

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

      2. distribute-neg-frac2 N/A

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

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

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

   3. Simplified 25.5%

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

   6. Applied egg-rr 29.4%

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

   7. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

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

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 14.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 14.0%

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

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 21.8%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 21.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 21.9%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]

   if 1.29999999999999998e-72 < C

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 37.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 37.0%

      \[\leadsto \frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}} \]
   10. Step-by-step derivation

       1. div-inv N/A

          \[\leadsto \frac{1}{\left(4 \cdot \left(A \cdot C\right) - B \cdot B\right) \cdot \color{blue}{\frac{1}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}}{\color{blue}{\frac{1}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\right), \color{blue}{\left(\frac{1}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}\right)}\right) \]

   11. Applied egg-rr 37.1%

       \[\leadsto \color{blue}{\frac{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B}}{{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)\right)}^{-0.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -6.5 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{A \cdot \left(4 \cdot C\right) - B \cdot B}}{{\left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;C \leq 9.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{1}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -4e+197)
   (/ (sqrt (* (* F -16.0) (* A (* A C)))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= C 9.6e-73)
     (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))
     (*
      (sqrt (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))))
      (/ 1.0 (- (* A (* 4.0 C)) (* B_m B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -4e+197) {
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 9.6e-73) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	} else {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (1.0 / ((A * (4.0 * C)) - (B_m * B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-4d+197)) then
        tmp = sqrt(((f * (-16.0d0)) * (a * (a * c)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (c <= 9.6d-73) then
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    else
        tmp = sqrt((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c)))) * (1.0d0 / ((a * (4.0d0 * c)) - (b_m * b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -4e+197) {
		tmp = Math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 9.6e-73) {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	} else {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (1.0 / ((A * (4.0 * C)) - (B_m * B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -4e+197:
		tmp = math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif C <= 9.6e-73:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	else:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (1.0 / ((A * (4.0 * C)) - (B_m * B_m)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -4e+197)
		tmp = Float64(sqrt(Float64(Float64(F * -16.0) * Float64(A * Float64(A * C)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (C <= 9.6e-73)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C)))) * Float64(1.0 / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -4e+197)
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (C <= 9.6e-73)
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	else
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (1.0 / ((A * (4.0 * C)) - (B_m * B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -4e+197], N[(N[Sqrt[N[(N[(F * -16.0), $MachinePrecision] * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 9.6e-73], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -3.9999999999999998e197

   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. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 2.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot A\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 27.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 27.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -3.9999999999999998e197 < C < 9.60000000000000022e-73

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 29.4%

      \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 14.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 14.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 21.8%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 21.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 21.9%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]

   if 9.60000000000000022e-73 < C

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 37.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 37.0%

      \[\leadsto \frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B} \cdot \color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\right), \color{blue}{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right), \left(\sqrt{\color{blue}{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(4 \cdot \left(A \cdot C\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\left(A \cdot C\right) \cdot 4\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot \left(2 \cdot 2\right)\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(\left(C \cdot 2\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(\left(2 \cdot C\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(\left(2 \cdot C\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(\left(C \cdot 2\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot \left(2 \cdot 2\right)\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \color{blue}{\left(2 \cdot F\right)}\right)}\right)\right) \]

       16. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4 \cdot 10^{+197}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;C \leq 9.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{1}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -1.26 \cdot 10^{+207}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -1.26e+207)
   (/ (sqrt (* (* F -16.0) (* A (* A C)))) (- (* (* 4.0 A) C) (* B_m B_m)))
   (if (<= C 5.5e-72)
     (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))
     (/
      (sqrt (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))))
      (- (* A (* 4.0 C)) (* B_m B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.26e+207) {
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 5.5e-72) {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	} else {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-1.26d+207)) then
        tmp = sqrt(((f * (-16.0d0)) * (a * (a * c)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else if (c <= 5.5d-72) then
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    else
        tmp = sqrt((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c)))) / ((a * (4.0d0 * c)) - (b_m * b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.26e+207) {
		tmp = Math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else if (C <= 5.5e-72) {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	} else {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -1.26e+207:
		tmp = math.sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m))
	elif C <= 5.5e-72:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	else:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -1.26e+207)
		tmp = Float64(sqrt(Float64(Float64(F * -16.0) * Float64(A * Float64(A * C)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	elseif (C <= 5.5e-72)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C)))) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -1.26e+207)
		tmp = sqrt(((F * -16.0) * (A * (A * C)))) / (((4.0 * A) * C) - (B_m * B_m));
	elseif (C <= 5.5e-72)
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	else
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) / ((A * (4.0 * C)) - (B_m * B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -1.26e+207], N[(N[Sqrt[N[(N[(F * -16.0), $MachinePrecision] * N[(A * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 5.5e-72], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.25999999999999999e207

   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. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 2.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left({A}^{2} \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left({A}^{2} \cdot C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 18.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), C\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot -16\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(A \cdot A\right) \cdot C\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(A \cdot C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \left(F \cdot -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 27.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 27.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -1.25999999999999999e207 < C < 5.49999999999999994e-72

   1. Initial program 22.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 25.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 29.4%

      \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 14.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 14.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 21.8%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 21.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 21.9%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]

   if 5.49999999999999994e-72 < C

   1. Initial program 20.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 37.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 37.0%

      \[\leadsto \frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}} \]
   10. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)}\right) \]

   11. Applied egg-rr 37.0%

       \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.26 \cdot 10^{+207}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot -16\right) \cdot \left(A \cdot \left(A \cdot C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{0.25}{A \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e-70)
   (*
    (sqrt (* (+ (* B_m B_m) (* A (* C -4.0))) (* F (* 4.0 C))))
    (/ 0.25 (* A C)))
   (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-70) {
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (0.25 / (A * C));
	} else {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 4.4d-70) then
        tmp = sqrt((((b_m * b_m) + (a * (c * (-4.0d0)))) * (f * (4.0d0 * c)))) * (0.25d0 / (a * c))
    else
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-70) {
		tmp = Math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (0.25 / (A * C));
	} else {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.4e-70:
		tmp = math.sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (0.25 / (A * C))
	else:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e-70)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(F * Float64(4.0 * C)))) * Float64(0.25 / Float64(A * C)));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.4e-70)
		tmp = sqrt((((B_m * B_m) + (A * (C * -4.0))) * (F * (4.0 * C)))) * (0.25 / (A * C));
	else
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e-70], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.25 / N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3999999999999998e-70

   1. Initial program 20.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 28.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot C\right)}, \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 18.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, C\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right)\right)\right) \]

   9. Simplified 18.5%

      \[\leadsto \frac{1}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B} \cdot \color{blue}{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\right), \color{blue}{\left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)}\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right), \left(\sqrt{\color{blue}{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(4 \cdot \left(A \cdot C\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\left(A \cdot C\right) \cdot 4\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(C \cdot \left(2 \cdot 2\right)\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(\left(C \cdot 2\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(A \cdot \left(\left(2 \cdot C\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(\left(2 \cdot C\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\color{blue}{\left(2 \cdot C\right)} \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(\left(C \cdot 2\right) \cdot 2\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot \left(2 \cdot 2\right)\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot \color{blue}{C}\right) \cdot \left(2 \cdot F\right)\right)}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \left(\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \color{blue}{\left(2 \cdot F\right)}\right)}\right)\right) \]

       16. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 18.5%

       \[\leadsto \color{blue}{\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}} \]

   12. Taylor expanded in A around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A \cdot C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, 4\right)\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, \left(A \cdot C\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), \mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, 4\right)\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 17.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, -4\right)\right)\right), \color{blue}{\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, 4\right)\right)}\right)\right)\right) \]

   14. Simplified 17.7%

       \[\leadsto \color{blue}{\frac{0.25}{A \cdot C}} \cdot \sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)} \]

   if 4.3999999999999998e-70 < B

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 28.9%

      \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 21.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 21.9%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 40.8%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 41.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 41.1%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)} \cdot \frac{0.25}{A \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F \cdot -16 + -4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{A \cdot A}}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1e-310)
   (* 0.25 (sqrt (/ (+ (* F -16.0) (* -4.0 (/ (* F (* B_m B_m)) (* A A)))) C)))
   (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = 0.25 * sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	} else {
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-1d-310)) then
        tmp = 0.25d0 * sqrt((((f * (-16.0d0)) + ((-4.0d0) * ((f * (b_m * b_m)) / (a * a)))) / c))
    else
        tmp = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = 0.25 * Math.sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	} else {
		tmp = -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1e-310:
		tmp = 0.25 * math.sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C))
	else:
		tmp = -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1e-310)
		tmp = Float64(0.25 * sqrt(Float64(Float64(Float64(F * -16.0) + Float64(-4.0 * Float64(Float64(F * Float64(B_m * B_m)) / Float64(A * A)))) / C)));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1e-310)
		tmp = 0.25 * sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	else
		tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1e-310], N[(0.25 * N[Sqrt[N[(N[(N[(F * -16.0), $MachinePrecision] + N[(-4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 35.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 10.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)}{A} + \frac{F \cdot \left(-2 \cdot \left(\left(B \cdot B\right) \cdot C\right) + \left(B \cdot B\right) \cdot \left(0 \cdot C\right)\right)}{A \cdot A}\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{\frac{-16 \cdot F + -4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\sqrt{\frac{-16 \cdot F + -4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\left(\frac{-16 \cdot F + -4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-16 \cdot F + -4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right), C\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot F\right), \left(-4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right), C\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \left(-4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right), C\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \left(\frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right)\right), C\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left(A \cdot A\right)\right)\right)\right), C\right)\right)\right) \]

      12. *-lowering-*.f64 36.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(A, A\right)\right)\right)\right), C\right)\right)\right) \]

   10. Simplified 36.5%

       \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{-16 \cdot F + -4 \cdot \frac{\left(B \cdot B\right) \cdot F}{A \cdot A}}{C}}} \]

   if -9.999999999999969e-311 < F

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 19.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 24.7%

      \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 11.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 11.3%

      \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

   10. Taylor expanded in A around 0

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 18.6%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
   13. Step-by-step derivation

       1. associate-*l/ N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

       5. sqrt-unprod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 18.7%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 18.7%

       \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F \cdot -16 + -4 \cdot \frac{F \cdot \left(B \cdot B\right)}{A \cdot A}}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{-1}{\frac{B\_m}{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ -1.0 (/ B_m (sqrt (* 2.0 (* B_m F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
}

Fortran

B_m = abs(b)
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 = (-1.0d0) / (b_m / sqrt((2.0d0 * (b_m * f))))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return -1.0 / (B_m / Math.sqrt((2.0 * (B_m * F))));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return -1.0 / (B_m / math.sqrt((2.0 * (B_m * F))))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(-1.0 / Float64(B_m / sqrt(Float64(2.0 * Float64(B_m * F)))))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -1.0 / (B_m / sqrt((2.0 * (B_m * F))));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(-1.0 / N[(B$95$m / N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

6. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

7. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 9.6%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 9.6%

   \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

10. Taylor expanded in A around 0

    \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
11. Step-by-step derivation

12. Simplified 16.4%

    \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
13. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right)\right) \]

    2. clear-num N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{B}{\sqrt{2} \cdot \sqrt{F \cdot B}}\right)\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right)\right)\right) \]

    5. sqrt-unprod N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right)\right)\right) \]

    6. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right)\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f64 16.5%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(B, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right)\right)\right) \]

14. Applied egg-rr 16.5%

    \[\leadsto -\color{blue}{\frac{1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}}} \]

15. Final simplification 16.5%

    \[\leadsto \frac{-1}{\frac{B}{\sqrt{2 \cdot \left(B \cdot F\right)}}} \]
16. Add Preprocessing

Alternative 21: 26.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{\sqrt{2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* 2.0 (* B_m F))) (- 0.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 * (B_m * F))) / (0.0 - B_m);
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 * (b_m * f))) / (0.0d0 - b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 * (B_m * F))) / (0.0 - B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((2.0 * (B_m * F))) / (0.0 - B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(2.0 * Float64(B_m * F))) / Float64(0.0 - B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (B_m * F))) / (0.0 - B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

6. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

7. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{2}}{B}\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \left(\sqrt{F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A + \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\left({A}^{2} + {B}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({A}^{2}\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(A \cdot A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left({B}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(B \cdot B\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 9.6%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{*.f64}\left(B, B\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Simplified 9.6%

   \[\leadsto \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{A \cdot A + B \cdot B}\right)}} \]

10. Taylor expanded in A around 0

    \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{B}\right)\right)\right)\right) \]
11. Step-by-step derivation

12. Simplified 16.4%

    \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \color{blue}{B}} \]
13. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot B}}{B}\right) \]

    2. distribute-neg-frac N/A

       \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)}{\color{blue}{B}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right), \color{blue}{B}\right) \]

    4. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot B}\right)\right), B\right) \]

    5. sqrt-unprod N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{2 \cdot \left(F \cdot B\right)}\right)\right), B\right) \]

    6. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot B\right)\right)\right)\right), B\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(F \cdot B\right)\right)\right)\right), B\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(B \cdot F\right)\right)\right)\right), B\right) \]

    9. *-lowering-*.f64 16.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(B, F\right)\right)\right)\right), B\right) \]

14. Applied egg-rr 16.5%

    \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{B}} \]

15. Final simplification 16.5%

    \[\leadsto \frac{\sqrt{2 \cdot \left(B \cdot F\right)}}{0 - B} \]
16. Add Preprocessing

Alternative 22: 5.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{-2 \cdot \sqrt{A \cdot F}}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (/ (* -2.0 (sqrt (* A F))) B_m))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return (-2.0 * sqrt((A * F))) / B_m;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = ((-2.0d0) * sqrt((a * f))) / b_m
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return (-2.0 * Math.sqrt((A * F))) / B_m;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return (-2.0 * math.sqrt((A * F))) / B_m

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(Float64(-2.0 * sqrt(Float64(A * F))) / B_m)
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = (-2.0 * sqrt((A * F))) / B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[(-2.0 * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]

Derivation

1. Initial program 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

5. Taylor expanded in A around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 7.6%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)}{A} + \frac{F \cdot \left(-2 \cdot \left(\left(B \cdot B\right) \cdot C\right) + \left(B \cdot B\right) \cdot \left(0 \cdot C\right)\right)}{A \cdot A}\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left(\sqrt{A \cdot F}\right), \color{blue}{\left(\frac{1}{B}\right)}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), \left(\frac{\color{blue}{1}}{B}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \left(\frac{1}{B}\right)\right)\right) \]

   5. /-lowering-/.f64 2.9%

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right)\right) \]

10. Simplified 2.9%

    \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \color{blue}{\frac{1}{B}} \]

    2. un-div-inv N/A

       \[\leadsto \frac{-2 \cdot \sqrt{A \cdot F}}{\color{blue}{B}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \sqrt{A \cdot F}\right), \color{blue}{B}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\sqrt{A \cdot F}\right)\right), B\right) \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right)\right), B\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(F \cdot A\right)\right)\right), B\right) \]

    7. *-lowering-*.f64 2.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right)\right), B\right) \]

12. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot A}}{B}} \]

13. Final simplification 2.9%

    \[\leadsto \frac{-2 \cdot \sqrt{A \cdot F}}{B} \]
14. Add Preprocessing

Alternative 23: 5.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{A \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
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);
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)
def code(A, B_m, C, F):
	return math.sqrt((A * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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 19.6%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Step-by-step derivation

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

3. Simplified 25.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
4. Add Preprocessing

5. Taylor expanded in A around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + 2 \cdot \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{*.f64}\left(2, \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(C + -1 \cdot C\right)\right) + 2 \cdot {B}^{2}\right)}{A} + \frac{F \cdot \left(-2 \cdot \left({B}^{2} \cdot C\right) + {B}^{2} \cdot \left(C + -1 \cdot C\right)\right)}{{A}^{2}}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

7. Simplified 7.6%

   \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) + 2 \cdot \left(\frac{F \cdot \left(-4 \cdot \left(C \cdot \left(0 \cdot C\right)\right) + 2 \cdot \left(B \cdot B\right)\right)}{A} + \frac{F \cdot \left(-2 \cdot \left(\left(B \cdot B\right) \cdot C\right) + \left(B \cdot B\right) \cdot \left(0 \cdot C\right)\right)}{A \cdot A}\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

8. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \color{blue}{\left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\left(\sqrt{A \cdot F}\right), \color{blue}{\left(\frac{1}{B}\right)}\right)\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), \left(\frac{\color{blue}{1}}{B}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \left(\frac{1}{B}\right)\right)\right) \]

   5. /-lowering-/.f64 2.9%

      \[\leadsto \mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \mathsf{/.f64}\left(1, \color{blue}{B}\right)\right)\right) \]

10. Simplified 2.9%

    \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(-2 \cdot \sqrt{A \cdot F}\right) \cdot \color{blue}{\frac{1}{B}} \]

    2. un-div-inv N/A

       \[\leadsto \frac{-2 \cdot \sqrt{A \cdot F}}{\color{blue}{B}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \sqrt{A \cdot F}\right), \color{blue}{B}\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \left(\sqrt{A \cdot F}\right)\right), B\right) \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right)\right), B\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\left(F \cdot A\right)\right)\right), B\right) \]

    7. *-lowering-*.f64 2.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right)\right), B\right) \]

12. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{\frac{-2 \cdot \sqrt{F \cdot A}}{B}} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\sqrt{F \cdot A} \cdot -2}{B} \]

    2. associate-/l* N/A

       \[\leadsto \sqrt{F \cdot A} \cdot \color{blue}{\frac{-2}{B}} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{F \cdot A}\right), \color{blue}{\left(\frac{-2}{B}\right)}\right) \]

    4. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot A\right)\right), \left(\frac{\color{blue}{-2}}{B}\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), \left(\frac{-2}{B}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \left(\frac{-2}{B}\right)\right) \]

    7. /-lowering-/.f64 2.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(A, F\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

14. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{-2}{B}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))